Stochastic modeling of the spot price of electricity incorporating

Transcrição

Stochastic modeling of the spot
price of electricity incorporating
commodities and renewables as
exogenous factors
DISSERTATION
zur Erlangung des Grades eines Doktors
der Naturwissenschaften
vorgelegt von
Jan Müller
eingereicht bei der Naturwissenschaftlich-Technischen Fakultät
der Universität Siegen
Siegen 2013
Gedruckt auf alterungsbeständigem, holz- und säurefreiem Papier.
GUTACHTER:
Prof. Dr. Alfred Müller, Universität Siegen
Prof. Dr. Rüdiger Kiesel, Universität Duisburg-Essen
TAG DER MÜNDLICHEN PRÜFUNG:
16. September 2013
Acknowledgements
First of all, I would like to thank Prof. Dr. Alfred Müller for his excellent
supervision of my PhD thesis and his support during the whole process of
research. He gave me the chance to pursue my research in various directions
and in personal responsibility.
Moreover, I am very grateful to Prof. Dr. Rüdiger Kiesel for his willingness
to accept the role as second referee.
I express my gratitude to the whole team of EnBW Trading GmbH for giving
me the opportunity to look at energy markets from a practitioner’s point of
view. Many fruitful discussions with my colleagues helped me to understand
the market mechanisms. Special thanks go to Dr. Markus Burger, Dr. Guido
Hirsch and Dr. Sven-Olaf Stoll for intensive discussions on modeling issues.
Their remarks and comments were appreciated contributions to this thesis.
V
Abstract
Models for electricity spot prices are used for a variety of valuation issues, e.g.
pricing of (real) options and pricing on the retail market. These issues require
adequate stochastic price models. A major requirement is the reproduction of
stylized facts of the spot price of electricity. Furthermore, the dependencies
on other commodities need to be modeled as well. Combined models of all
relevant commodities gain importance in the context of risk management of
energy utilities. The complexity of options and of portfolios of energy utilities
increases due to the number of products with multiple commodities included.
This makes the use of combined models essential.
The spot price of electricity is set by the principle of merit order. The most
important drivers for the merit order curve on the EEX market are grid load,
generation of renewables and prices of coal, natural gas, oil and emission allowances. We present a model incorporating these influence factors in a functional approximation of the non-deterministic merit order curve. The stylized
facts mean reversion, seasonalities, negative prices and price spikes as observable in historical spot prices are reproduced.
The input factors require models as well. The residual load model relies on
distinct models for renewables including their increase and grid load. We introduce a gas price model incorporating temperature and oil price as exogenous
factors. Three model alternatives are presented for the considered commodity
prices. A comparison gives evidence that the cointegration approach describes
the dependencies best. Finally, the electricity price model is applied to riskadequate pricing on the retail market.
VII
Zusammenfassung
Für eine Reihe von Bewertungen werden Strompreismodelle benötigt, beispielsweise für die Bewertung von (Real-) Optionen oder die Risikobewertung bei
Kundenverträgen. Diese Anwendungen erfordern stochastische Spotpreismodelle, die die Eigenschaften der Preise sowie die Abhängigkeiten von anderen
Brennstoffpreisen wiedergeben. Für das Risikomanagement von Energieversorgern sind kombinierte Preismodelle für alle Brennstoffe und CO2 von großer
Bedeutung. Durch die steigende Zahl von Handelsprodukten, die auf mehreren
Brennstoffpreisen basieren, steigt die Komplexität der zu bewertenden Portfolios. Daher sind kombinierte Modelle unverzichtbar.
Der Stromspotpreis wird durch das Prinzip der "merit order" bestimmt. Dabei
sind die Netzlast, die Erzeugung durch erneuerbare Energien und die Preise
für Kohle, Gas, Öl und CO2 -Zertifikate die wichtigsten Einflussfaktoren auf
dem EEX Markt. In dieser Arbeit wird eine funktionale Approximation der
nicht-deterministischen "merit order" unter Berücksichtigung der wesentlichen
Einflussfaktoren angegeben. Das resultierende Modell reproduziert mean reversion, Saisonalitäten, negative Preise und Preisspitzen der historischen Preise.
Für die Einflussfaktoren des Strompreismodells sind ebenfalls Modelle notwendig. Das Restlastmodell basiert auf Modellen für die Netzlast und die
Erzeugung der erneuerbaren Energien unter Berücksichtigung der erwarteten
Kapazitätserhöhungen. Im Gaspreismodell werden Temperatur und Ölpreis als
exogene Größen verwendet. Für die Brennstoffpreise werden drei verschiedene
Modellansätze präsentiert. Der Vergleich dieser Ansätze zeigt, dass die Abhängigkeiten im Kointegrationsansatz am besten berücksichtigt werden. Abschließend wird das Modell zur Bewertung der Risiken in Kundenverträgen
angewendet.
IX
Contents
List of symbols
XIII
1. Introduction
1
2. Commodity markets
2.1. Electricity market
2.2. Coal market . . .
2.3. Gas market . . .
2.4. Oil market . . . .
2.5. Emissions market
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3. Modeling the spot price of electricity
3.1. The merit order curve . . . . . . . . . . . .
3.2. Literature on electricity price models . . . .
3.3. The new spot price model . . . . . . . . . .
3.3.1. The polynomial approximation . . .
3.3.2. The stochastic process . . . . . . . .
3.4. Modeling the load . . . . . . . . . . . . . . .
3.4.1. Modeling the grid load and renewable
3.4.2. Modeling the residual load . . . . . .
3.5. Inclusion of futures market information . . .
.
.
.
.
.
5
5
8
11
13
16
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
energy generation
. . . . . . . . . . .
. . . . . . . . . . .
19
19
26
28
30
40
43
44
53
55
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4. Modeling the spot price of natural gas
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.2. A review of the model by Stoll and Wiebauer (2010)
4.3. The oil price dependence of gas prices . . . . . . . . .
4.4. Model calibration with temperature and oil price . .
4.4.1. Oil price model . . . . . . . . . . . . . . . . .
4.4.2. Temperature model . . . . . . . . . . . . . . .
4.4.3. The residual stochastic process . . . . . . . .
4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
57
57
59
62
64
67
69
69
70
5. Modeling commodity prices
73
5.1. Literature on commodity price models . . . . . . . . . . . . . . 74
5.2. Three-dimensional random walk . . . . . . . . . . . . . . . . . . 75
XI
Contents
5.3. Correlated two factor model . . . . . . . . . . . .
5.4. Cointegration . . . . . . . . . . . . . . . . . . . .
5.4.1. Definitions and properties of cointegration
5.4.2. Parameter estimation and testing . . . . .
5.4.3. Application to commodity prices . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
76
82
83
86
89
6. Discussion of electricity price models
93
6.1. Overview of presented models . . . . . . . . . . . . . . . . . . . 93
6.2. Analysis of model features . . . . . . . . . . . . . . . . . . . . . 95
6.3. Out-of-sample model validation . . . . . . . . . . . . . . . . . . 98
7. Pricing of retail power contracts
7.1. Introduction . . . . . . . . . . . . . . .
7.2. Price components . . . . . . . . . . . .
7.2.1. Hourly Price Forward Curve . .
7.2.2. Risk factors . . . . . . . . . . .
7.2.3. Risk-adequate return on capital
7.3. Correlated price-load models . . . . . .
7.3.1. A customer load model . . . . .
7.3.2. Example . . . . . . . . . . . . .
7.4. Conclusion . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8. Conclusion
A. Mathematical appendix
A.1. Stochastic processes . . . . . . . . . . . . . . . . .
A.1.1. Definitions and properties . . . . . . . . .
A.1.2. Parameter estimation of ARMA processes
A.1.3. Test for stationarity . . . . . . . . . . . .
A.2. Bessel function . . . . . . . . . . . . . . . . . . .
A.3. Probability distributions . . . . . . . . . . . . . .
A.3.1. Student’s t-distribution . . . . . . . . . . .
A.3.2. Inverse Gaussian distribution . . . . . . .
A.3.3. Gamma distribution . . . . . . . . . . . .
A.3.4. Generalized inverse Gaussian distribution .
A.3.5. Generalized hyperbolic distributions . . . .
B. Programming
Bibliography
XII
103
. 103
. 106
. 106
. 107
. 112
. 113
. 113
. 116
. 118
121
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
XV
. XV
. XV
. XVIII
. XIX
. XXI
. XXI
. XXII
. XXII
. XXII
. XXIII
. XXIII
XXVII
XXIX
List of symbols
Chapter 3 - Modeling the spot price of electricity
1A (t)
Indicator function
bt
X
Forecast of a time series Xt
Bt
Hourly power generation by biomass and minor energy sources
Ct
Daily price of coal in Euro per ton
Et
Daily price of emission allowances in Euro per ton
g
Polynomial approximation of the merit order curve
Gt
Daily day-ahead price of natural gas in Euro per megawatt hour
L
Lag operator
Lt
Hourly grid load
lt
Hourly residual load
Nt
Hourly power generation by nuclear energy
Ot
Daily price of crude oil in Euro per barrel
P Vt
Hourly power generation by solar energy
St
Hourly day-ahead price of electricity in Euro per megawatt hour
vt
Hourly average availability of power plants
Wt
Hourly power generation by wind energy
W N (0, σ 2 )
White noise process with mean 0 and variance σ 2
(Z)
Stochastic short term process belonging to the model for Zt
(Z)
Stochastic long term process belonging to the model for Zt
Xt
Yt
XIII
Contents
Chapter 4 - Modeling the spot price of natural gas
Λd,w
Normalized cumulated heating degree days on day d of winter
w, also denoted as Λt
Ψt
Oil price formula
CHDDd,w
Cumulated heating degree days on day d of winter w
HDDd,w
Heating degree days on day d of winter w, also denoted as
HDDt
Tt
Daily average temperature in ◦ C
Chapter 5 - Modeling commodity prices
β
Cointegrating vector
Γi
Coefficient matrix of the VAR(k) process in VECM representation
Ω
Covariance matrix
Πi
Coefficient matrix of the VAR(k) process
χt
Short term process of the two factor model
∆
Difference operator
NN (0, Ω)
N-dimensional normal distribution with mean vector 0 and covariance matrix Ω
ρ
Correlation of short term and long term process of the two
factor model
b
a
Estimator of a parameter a
ξt
Long term process of the two factor model
A(z)
Characteristic polynomial of the VAR(k) process
H(r)
Cointegration model of rank r
In
(n × n) identity matrix
r
Rank of cointegration
XIV
1. Introduction
Electricity markets started as local markets where the generation from any
power plant was consumed by surrounding companies and households. Due to
the increase of transmission networks and the market liberalization in the late
90’s local electricity markets developed to national or international markets
with competing suppliers. Instead of relying on the local supplier customers
began to choose between several suppliers. Generation capacities might exceed
the demand for some supplier and vice versa. In general terms, generation and
consumption in a supplier’s portfolio might not match anymore. On the one
hand, suppliers with surplus generation try to sell their power in order to
maximize their profit. On the other hand, suppliers with a lack of generation
buy electricity to meet their customers’ demand. Thus, competition makes
suppliers start trading electricity.
A portfolio of customers requires the allocation of strongly varying quantities of
electricity at any time. This flexibility might exceed the flexibility of generation
assets so that the supplier needs to purchase additional flexibility from the
market. In addition to the physical need for electricity, there are more reasons
for trading electricity. Future contracts are used to hedge generation and
demand of energy utilities. Furthermore, speculators trade future contracts.
Energy exchanges, such as the European Energy Exchange (EEX) in Leipzig,
offer a variety of trading products. While electricity with hourly delivery for
the following day is traded in the spot market, contracts with longer future
delivery periods such as quarters or years are traded in the futures market.
In addition to these standard trading products, a wide range of derivatives,
partly known from financial markets, is traded in the electricity market: plain
vanilla options, multi-exercise options or virtual power plants. These financial
instruments increase a supplier’s flexibility.
Derivatives in the electricity market need to be priced as any products in financial or energy markets. In some cases, e.g. plain vanilla options on future
contracts, well-known concepts from financial markets can be adapted to derive
prices. Most electricity derivatives are specific for the energy market. Therefore, an adaptation of concepts from financial markets is not always possible.
1
1. Introduction
For example, swing options contain a certain number of exercise rights within
a certain period of time. Such complex options are not easy to price as optimal exercise strategies are required. Analytic pricing formulas do not exist but
Monte Carlo methods are commonly used. Pricing of real-options like power
plants or risk-adequate pricing of retail power contracts are further examples
where Monte Carlo methods are applied. These examples require simulation
models for the underlying electricity prices. Derivatives on electricity can be
distinguished with respect to the underlying: prices of future contracts or spot
prices. As many derivatives such as multi-exercise options and virtual power
plants are based on the spot price, this work focuses on spot price models for
electricity.
At first sight, electricity prices are the result of supply meeting demand. As
large amounts of electricity generation are not storable at reasonable costs supply may not exceed demand. Thus, the price is set by the demand. This idea
is the basis for many spot price models in the literature where the load is used
as a fundamental factor. A variety of functional dependencies between load
and spot price is proposed. Such models neglect several important influence
factors:
1. Fuel prices.
2. Prices of emission allowances.
3. Generation of renewables.
Electricity cannot be generated at the same price at different times due to
varying costs of generation. This is due to fluctuations of generation capacities and volatility of commodity prices. For example, coal-fired power plants
constitute a considerable part of generation capacities in many markets. Thus,
the price of electricity is affected by new coal-fired power plants as well as by
changes of prices of coal and emission allowances. The former makes the new
plant replace more expensive power plants on the merit order curve. Volatility
of commodity prices has a direct influence on generation costs of each power
plant. Thus, increased commodity prices directly lead to increased electricity
prices. Both influences need to be covered by an adequate spot price model.
In this work a new stochastic model for the spot price of electricity is introduced. This model accounts for fluctuations of demand, commodity prices,
renewables and capacities. To our knowledge there is no model in the literature
combining all these influence factors in one model framework. We introduce
a function of residual load and prices of coal, gas and emission allowances to
explain the spot price behavior. Especially, the inclusion of commodity prices
2
increases the variety of possible applications of our model. The valuation of
thermal power plants relies on models covering electricity prices as well as
prices of the underlying fuel and prices of emission allowances. Derivatives
on electricity or any of the commodity prices mentioned above can be priced
by our model. This includes load-dependent products such as full-service contracts in the retail market.
Price formation in the EEX market is the result of an auction where bids and
offers of energy utilities determine the price. Within this auction the so-called
merit order curve is constructed. A function of residual load and commodity
prices in our model is an approximation of the merit order curve. Deviations
from the curve and model errors are covered by a stationary stochastic process.
Apart from the deduction of the spot price model we introduce models for the
underlying input variables.
The residual load is composed of grid load, nuclear generation and generation of
renewables. A grid load model including seasonalities and a long term factor for
economic influences is proposed. This component induces seasonalities to the
spot price model. Recently, the spot price behavior changed due to the strong
increase of renewables and the decrease of nuclear capacities. Both changes
and expectations about future capacities are regarded in the corresponding
models.
In addition to the residual load model we present models for the prices of gas,
coal, oil and emission allowances. A gas price model incorporating temperature and oil price as exogenous factors is introduced. It is shown that these
fundamental components are important drivers of the gas price. The oil price is
part of a combined model with prices of coal and emission allowances. Due to
this model framework correlations between prices of gas, coal, oil and emission
allowances are incorporated in the electricity price model.
As an alternative approach we present a cointegrated model for the four commodities considered within this work. This concept describes a stronger relationship than correlation between these variables. Evidence for this alternative is given by an out-of-sample validation using the clean dark spread. This
spread is the basis for the valuation of coal-fired power plants being one of the
typical applications of a multi-commodity model. Apart from valuations of
further (real) options on the wholesale markets, there are applications for the
model in the retail market. We present risk factors for suppliers participating
in the retail market including a model framework to derive prices for typical
retail power contracts. As a spot price model including the grid load is essential within this framework we can apply our new model. The application to a
retail pricing example gives further evidence for the reliability of our models.
3
1. Introduction
Structure of the thesis
After this introduction we give a short overview of the commodity markets
that are relevant within this thesis. We restrict ourselves to the information
necessary to understand the following modeling issues. In Chapter 3 our spot
price models are introduced. We describe the price formation on the EEX
market and derive adequate spot price models. These are categorized into the
existing literature. The underlying models for the residual load are presented
as well. The gas price model is presented in Chapter 4. Different modeling
approaches for commodity prices are presented in Chapter 5.
After introduction of all models their features are compared in Chapter 6. This
includes an out-of-sample model validation. Chapter 7 contains the application
of the model to the issue of risk-adequate pricing of retail power contracts. The
results of the thesis are concluded in Chapter 8.
4
Modeling of commodity prices requires knowledge about the market structures
and the traded products. We give a short overview of the relevant markets
for our models. This includes the choice of adequate data to estimate the
model parameters. For further details on commodity markets we refer to more
comprehensive publications like Burger et al. (2007) and Geman (2005).
There are two characteristics that distinguish electricity from other commodities:
1. Electricity is not storable to a large extent. The generation exceeds the
capacities of (hydro pumped) storages by far.
2. Transport of electricity is linked to transmission networks.
These facts have consequences for the market structure, price behavior and
the products.
Which market is considered? Due to transmission networks the electricity
market is not global. Adjacent national electricity markets start to link to
one another but the capacities give natural constraints. Therefore, we focus
on the presentation of the German electricity market which is relevant for our
modeling. The trade of standardized contracts in Germany takes place on the
European Energy Exchange (EEX) in Leipzig (see European Energy Exchange
(2013)). In addition to the trade on the exchange, there is an over-the-counter
(OTC) market for any kind of contracts. The standard products are traded
OTC as well as unusual specifications like peakload on weekends.
5
Where are the producers? Various energy utilities operate nuclear, coalfired, gas-fired and petrol-fired power plants. A wide range of renewable energy sources owned by energy utilites, industrial companies or even private
households contributes to the total generation as well.
What products are traded? A major part of the electricity trading takes
place in the futures market. Future contracts describe delivery of electricity
during a certain period of time (year, quarter, month, week). A further distinction of future contracts is given by the delivery hours. So-called baseload
contracts guarantee delivery of energy in every hour of the period. In contrast, delivery of peakload contracts is restricted to 8 a.m. till 8 p.m. on
working days. Both contract types are used for supply (physical) or hedging (financial) purposes. These are standard products liquidly traded on the
EEX.
In addition to the futures market there is a day-ahead market. Everyday
the EEX determines the price of electricity for every hour of the following
day in an auction. The power plant operators submit quotes, i.e. a list with
available capacities and corresponding prices, and the potential buyers submit
their bids. The EEX publishes those 24 prices that lead to a match of bids
and offers in every hour. More details on the formation of prices are given in
Section 3.1. In the following the prices on the day-ahead market are referred
to as spot prices.
Who are the traders? Market participants on the exchange and OTC market can be differentiated into three groups:
• Energy utilities trade for financial and physical purposes.
• Banks participate in the markets for speculation.
• Some major industrial companies have market access to buy electricity
for their own use.
Due to trading volumes the price is set by energy utilities and banks. Their
focus is on liquidly traded products like the baseload contract for the front
year. Front year describes the first yearly contract being traded. E.g. the
front year contract in July 2012 would be the future contract with delivery in
2013. The corresponding terms front month and front quarter are used as well.
Yearly contracts for the following years are less liquidly traded but important
6
for energy utilities hedging their generation of electricity.
Which data is used? Recent future prices are needed to include all available market information into our spot price models. Market participants might
have knowledge about price relevant information that cannot be deduced from
historical spot prices. Therefore, we include future prices as described in Section 3.5. The historical prices of the future contracts for 2013 are given in
Figure 2.1.
Figure 2.1.: Prices of baseload and peakload contracts for the year 2013 as
quoted on the EEX from January 2007 till September 2012.
The most relevant data in this work concerns the spot price of electricity. We
use the prices on the EEX for parameter estimations of the model. Amongst
others, the minimum price of -500.02 EUR/MWh at 2 a.m. on 2009/10/04.
is cut off to ensure that some structures can be observed in Figure 2.2. The
contrast between spot prices and the prices of future contracts seen in Figure
2.1 is due to delivery periods: The spot prices are prices for delivery in a single
hour of the following day. The prices of future contracts describe delivery
within the year 2013. This means that any kind of seasonality within the year
is removed by considering an average price for the year. As future prices are
average prices for a delivery period they usually do not exhibit extreme prices
7
as spot prices do. Extreme spot prices occur in various situations such as a
combination of low load and high infeed from renewable energy (low prices) or
power plant outages (high prices). Examples of typical price patterns within a
Figure 2.2.: Spot prices of electricity on the EEX from January 2009 till
September 2012. Prices below -100 EUR/MWh are cut.
week are given in Figure 2.3. Prices at night and on weekends are remarkably
lower than at daytime or on weekdays. Further typical price patterns such
as maximal prices at noon have changed due to the increase of generation by
renewables (compare Section 3.1).
2.2. Coal market
The market for black coal is not a local or regional but a global market. This
is due to the resources and the ports all over the world. As lignite plays a
minor role within this work the term coal always refers to black coal.
Which market is considered? The German electricity prices are influenced
by prices of coal in Germany. As the German coal production decreased dur-
8
2.2. Coal market
Figure 2.3.: Spot prices of electricity on the EEX from 2009/08/17 till
2009/08/23 (blue line) and from 2012/08/13 till 2012/08/19
(green line).
ing the last decades the coal prices in Germany are set by coal imported from
abroad. Coal prices on the EEX and coal prices on the IntercontinentalExchange (ICE), London, (see IntercontinentalExchange (2013)) both refer to
imported coal. The prices on the ICE are more reliable due to higher trading
volumes. Both exchanges offer various standardized products with future delivery. A spot market meaning immediate delivery does not exist due to the
transportation time of coal.
Where are the producers? The major producers are China, United States,
India, Russia, Australia, South Africa, Indonesia, Poland, Kazakhstan, Colombia and Ukraine. Due to their own consumption and freight charges not all of
these countries export coal to any other country in the world. The most important coal producers for Germany are Colombia, Russia (and the countries
of the former Soviet Union), United States and South Africa.
What products are traded? As coal is a heterogenous commodity there
are different qualities. These differences result from varying carbon, energy,
9
sulphur and ash content. These contents influence caloric value and emissions.
As a consequence the price of coal is dependent on the quality.
Another influence factor for the price is given by the Incoterms (international
commerce terms). These terms specify the allocation of costs concerning the
transport:
• Free on board (FOB): The seller delivers to the port of shipment and
pays the loading costs. As soon as the goods are on the ship the buyer
bears all other costs.
• Cost, insurance and freight (CIF): All costs of transportation and
insurance are included in the selling price.
Varying quality, transportation costs and Incoterms lead to distinct prices for
coal. Therefore, physical coal trading is done by OTC deals. The API#2
index is the average price of a certain coal quality being delivered CIF ARA
(Amsterdam, Rotterdam, Antwerp) within the next 90 days. Although it is
not guaranteed that all coal deals are captured by the index it is the most
important benchmark for physical coal trades in Central Europe. Further
price indexes for different regions, different qualities and different Incoterm
specifications exist but are not relevant in this work.
In addition to the physical trade there is an extensive trade of financial coal
contracts on the ICE. Contracts for months, quarters, seasons (summer and
winter) and years are traded. The API#2 index describing delivery within the
next 90 days is the closest to spot. But for reasons of liquidity we use prices
of the front month contract traded on the ICE instead of the API#2 index
prices.
Who are the traders? There are four major groups of market participants
in the exchanges and OTC that influence the price:
• Energy utilities trade for financial and physical purposes.
• Banks participate in the markets for speculation.
• Metallurgical industrial companies need coal for their production
processes.
• Coal producers sell their production.
10
2.3. Gas market
Which data is used? All coal prices are quoted in US dollars. As we analyze
the coal price effect on the electricity price quoted in Euro coal prices in Euro
per ton are used. Forward exchange rates are used for the conversion. The
price data consisting of five different coal futures traded on the ICE is shown
in Figure 2.4. For the electricity price model the front month prices of coal
are used as an input variable. All futures are required to set up the coal price
model in Section 5.3.
Figure 2.4.: Prices of coal futures with different maturities from January 2009
till September 2012.
2.3. Gas market
The trade of natural gas developed later than the trade of coal but due to
the increasing importance of gas as a fuel for heating purposes the market is
rapidly growing.
Which market is considered? The major way of transportation is a gas
pipeline. Therefore, the market for gas is a regional one. The German gas
11
market was split into several market areas. After some steps of consolidation
the two market areas Gaspool and NetConnect Germany (NCG) are left. The
Dutch market is covered by the trading hub Title Transfer Facility (TTF).
The Dutch gas market started earlier than the German and leads to a longer
history of price data. The high pipeline capacities between both countries
eliminate arbitrage possibilities. Both markets consist of various standardized trading products in the spot and future market. In Germany the EEX
allows for trading in both market areas. The European Energy Derivatives Exchange (ENDEX), Amsterdam, is the exchange for TTF (see European Energy
Derivatives Exchange (2013)).
Where are the producers? The main sources of natural gas in the Western
European gas markets are Russia, Norway and the Netherlands. Especially the
German gas market relies on imported gas from these countries. The increasing
number of liquefied natural gas (LNG) transports to Central Europe increase
the available quantities as well as the variety of sources. LNG transports,
e.g. from Qatar, arrive at British or Dutch LNG terminals and influence the
German market as well.
What products are traded? Spot and forward products are liquidly traded
in both markets. Buying a forward contract on gas means delivery of a certain
quantity during the entire delivery period. Standardized delivery periods on
exchanges are months, quarters, years and seasons (October till March, April
till September). The so-called spot market is a day-ahead market. The "spot"
price quoted on one day is a price for delivery of gas on the next business day.
In this context the day starts at 6 a.m. and ends at 6 a.m. on the following
day. Another specialty concerns the weekends: A weekend price is quoted in
addition to the day-ahead price. The weekend price is the price for delivery
of gas on Saturday and Sunday. As a consequence the day-ahead price quoted
on Friday describes the price for delivery on Monday.
Who are the traders? Only two groups trade in the spot market for gas:
• Energy utilities supply their customers and operate gas-fired power
plants and gas storages.
• Some major industrial companies have access to the market to buy
gas for their own use.
12
2.4. Oil market
Which data is used? The spot price of natural gas is used to explain the
spot price of electricity. As the ENDEX provides a longer consistent history
of spot prices we choose this price data. As argued above the price differences
are negligible. Using day-ahead prices for weekdays and weekend prices for
Saturday and Sunday a daily time series of spot prices can be constructed.
This data is shown in Figure 2.5. Extreme spot prices can occur in analogy
to power spot prices. For example, technical problems with gas storages in
winter immediately increase the price.
Figure 2.5.: Spot prices of gas on TTF from October 2004 till September
2012.
Prices of future contracts are needed for the combined commodity price model
in Section 5.4. The time series of front month prices for the TTF market area is
given in Figure 2.6. This data contains the decreasing prices as a consequence
of the crisis 2008/2009.
2.4. Oil market
Throughout this work oil is the abbreviation for crude oil. Reasons why gasoil
or fueloil are not used are given in the particular chapters.
13
Figure 2.6.: Prices of front month contracts of gas on TTF from January
2009 till September 2012.
Which market is considered? The oil market is not a regional but a global
market. The crude oil grade Brent is the most relevant for the European
market. Liquid trading of future contracts takes place on the ICE. Physical
trades are OTC and include an oil price as well as costs for transportation. As
these costs differ there is no standardized trade of physical oil shipments on
an exchange.
Where are the producers? Major oil reserves can be found in the Middle
East, the former Soviet Union and North America. Countries without any
considerable oil reserves, like Germany, rely on supply from these regions (or
other oil producers not mentioned here).
What products are traded? Similar to coal, oil has different qualities depending on the source. The most important characteristics are viscosity and
sulphur content. In order to achieve some standardization for trading purposes
some oil qualities became benchmarks. The relevant price index for oil in Germany is the Brent index. This index describes the North Sea oil market with
crude oil having 0.3% sulphur content and 38◦ API viscosity. Although the
14
2.4. Oil market
oil reserves in the Brent area declined the price index is still used in Europe.
The exact price of an oil shipment is determined by the transport costs, the
Incoterms and the oil price itself.
Monthly financial future contracts are liquidly traded on the ICE and OTC.
Price quotes for the upcoming 36 front month are available. Physical oil trades
are OTC deals. Their prices are closely related to the exchange prices for
the corresponding period of time. The energy news service Platts (see Platts
(2013)) assesses prices for physical Brent shipments loading FOB 10-25 days
forward. These prices are known as dated Brent. Spot prices as known from
other markets do not exist due to the transportation by ship.
Who are the traders? The trading volumes in the financial oil market exceed
the physically traded volumes by far. Financial trades take place for reasons
of speculation or hedging of physical demand.
• Banks and funds trade for speculation.
• Energy utilities need to supply their power plants with oil and hedge
these prices. Speculation is another purpose of them.
• The highest physical demand have refineries.
• The chemical industry needs oil as the basis for many processes.
Which data is used? As prices of financial future contracts for the front
month and prices of dated Brent nearly refer to the same period of time we
choose the more liquid prices of future contracts for our model calibration.
Another reason is the easier availability as prices quoted on the ICE are public
information.
As we use historical data for the parameter estimation in our model we need
to make sure that currencies of electricity, gas and oil prices are consistent. To
exclude any future currency effects Brent prices are converted from Dollar per
Barrel into Euro per Barrel by means of forward exchange rates. Though the
spot price model for electricity relies only on the front month contract we need
prices of Brent futures with five different delivery months (see Figure 2.7) for
the commodity price model.
15
Figure 2.7.: Prices of Brent futures with different maturities from January
In order to reduce emissions the European Union decided that companies in
industries with high carbon dioxide emissions need allowances for their emissions. Therefore, the European Union started the European Union Emission
Trading System (EU ETS) in 2005. The EU ETS consists of trading periods.
The first period from 2005 to 2007 and the second period from 2008 to 2012
started with allocation of allowances to companies that cause emissions. As
these allocations did not exactly match the realized demand for allowances the
companies started trading. By the end of April allowances for the emissions of
the year before need to be submitted. In the third period from 2013 to 2020
the allocation changes and companies need to buy all allowances. Nevertheless,
a sufficient number of allowances can be achieved by means of trading. A lack
of allowances causes high penalties. For more detailed information including
the official documents see European Commission (2013).
Which market is considered? Emission allowances are traded on the EEX
and ICE. Due to its size and the relevance for Europe the trade on the ICE
16
is more liquid than on the EEX. In addition, an extensive OTC trade takes
place. There is no arbitrage between prices on the ICE and prices on the EEX
or OTC.
Where are the producers? Allowances are allocated to certain companies
owning plants with high emissions. These companies can sell surplus allowances on the market.
What products are traded? One contract for the upcoming months March,
June, September and December is quoted on the ICE. Contracts with delivery
in December are quoted for several years. In contrast to commodities such
as electricity or gas, allowances can be delivered at one point of time. The
product specifications on the ICE determine the day of delivery as the last
Monday of the month being a business day. If there are less than four business
days following the last Monday, the expiry will be on the penultimate Monday.
Spot trade of allowances with immediate delivery is possible in case of emission
allowances but it only takes place OTC.
Who are the traders? The market participants on the market for emission
allowances have different trading purposes.
• All industrial companies emitting CO2 sell surplus allocated emission
allowances or buy additional allowances.
• Energy utilities hedge their future emissions due to planned electricity
generation by their power plants.
• Banks and financial institutions speculate on prices of emission allowances.
Which data is used? The number of allocated allowances in period one
exceeded the demand. The companies realized this situation at the end of
the period and sold their surplus of allowances. As a consequence the price
declined close to zero. At the beginning of period two, the price raised to the
usual level. This price effect is due to the regulations on the emissions market.
It is not a result of the typical price behavior. Therefore, we exclude period
one from our data.
17
Although prices for delivery in various months are quoted, the trade focusses on
the December contract of the current year. Therefore, we use the price of the
December contract for the current year as the basis for our model calibration.
The March contract is important as a last chance to get allowances for the
previous year. Nevertheless, due to speculation the trading volumes of the
December contracts are higher.
In contrast to the other commodities mentioned above we do not need prices
close to spot in case of emission allowances. The allowances need to match
the emissions by the end of a year. Therefore, there is no difference between
buying allowances on the spot market, in December of the same year or in
March of the following year. Thus we use the most liquid prices for model
calibration, i.e. the prices of the December contract on the ICE. The price
data used within this work is shown in Figure 2.8. Several future contracts are
used to calibrate the combined commodity price model presented in Section
5.3.
Figure 2.8.: Prices of emission allowances with different maturities from January 2009 till September 2012.
18
3. Modeling the spot price of
electricity
There are various factors with a considerable influence on the spot price of
electricity. These factors and their effect on the price behavior are presented in
this chapter. Requirements for adequate spot price models can be derived from
this list of features. The wide range of existing literature on price models is
distinguished with respect to their focus on some of the price features. Finally,
we introduce a new model meeting the requirements. As the spot price model
depends on load and renewables as input factor we present the corresponding
models. A short overview of the model, its alternative and the underlying load
model can be found in Section 6.1.
3.1. The merit order curve
The generation of electricity in Germany relies on various energy sources (see
Figure 3.1). Traditionally the most important sources were lignite, coal, gas
Figure 3.1.: Classification of major energy sources.
and uranium. Within the last years the generation portfolio experienced large
changes. Capacities of renewable energy sources, especially wind and photovoltaic (PV) power plants, extremely increased. In contrast the capacities of
19
nuclear power plants were reduced by the German government as a consequence
of the nuclear disaster in Fukushima, Japan, in March 2011. The generation
volumes distinguished by energy sources in 2009 and 2011 are given in Figure
3.2 (see Statistisches Bundesamt (2013)). The obvious decrease of electricity
generation by uranium is covered by an increase in biomass (major part of
"Others"), wind and PV generation.
Figure 3.2.: Comparison of percentages of generated electricity by energy
sources in Germany for 2009 and 2011. Total generation: 592.4
TWh in 2009 and 614.5 TWh in 2011.
Pricing on the EEX
The total demand for electricity needs to be met by generated electricity from
various energy sources and imported quantities. The relation between demand
and generation plus import determines the price. The price is determined in
a standardized process on the EEX. All market participants submit bids and
offers for arbitrary quantities. The bids and offers usually include a maximum
buying price and a minimal selling price. For example, the use of various
buying price limits might ensure that lower quantities are bought at higher
prices. This might be reasonable if a market participant has some expensive
20
generation capacities to match the own demand. These capacities will only
be used above a certain price on the exchange. All orders in the auction are
restricted to the price range from -3000 EUR/MWh to 3000 EUR/MWh. The
auction takes place every day at 12:00 p.m. for every hour of the following
day. 24 order sets need to be submitted by the market participants.
Assuming a market participant with a coal-fired power plant tries to sell its
generation on the spot market. Due to the auction structure the result of
the auction might be that the plant runs only one hour on that day. But
this would not be profitable as there are high starting costs for such power
plants. To avoid such problems block orders are allowed in the auction as
well. The market participants may place orders for blocks of several adjacent
hours instead of single hours. These orders are either completely accepted
or completely rejected. This situation allows for the profitable marketing of
power plants in the spot market.
The exchange constructs a so-called merit order curve from all incoming
offers for every hour. The offers are sorted ascending with respect to the
prices. The corresponding quantities are cumulated. As the offers are based
on power plants with costs that strongly depend on the fuel price the sorted
curve consists of several blocks. These blocks can usually be dedicated to the
different fuel types. An overlapping of adjacent blocks is possible. New plants
of an expensive fuel with a high efficiency might sometimes be cheaper than
old plants of a cheaper fuel type with a low efficiency. Therefore, not all plants
of one fuel type produce with the same variable costs. Hence, the offers will
be different as well. Variable costs resulting from fuel prices are increased by
further variable costs and fixed costs of the power plant.
Variable costs of a power plant do not only depend on fuel costs but also on
(high) costs for starting the plant. E.g. coal plants need to be preheated
which is usually done by burning petrol. In order to avoid these starting costs
power plants sometimes produce and sell electricity below fuel costs. For these
reasons, the merit order curve is not constructed by sorting the fuel prices and
adding up the capacities of the corresponding fuel types. The typical order
of power plants (depending on prices) would be nuclear, lignite, coal, gas (as
a general term for gas used in gas turbines and combined cycle gas turbines)
and oil. Two example of the merit order curve on the EEX are given in Figure
3.3. Small price movements might change place of single power plants within
the merit order curve. Extreme price movements might even change places of
all power plants of different fuels. For this reason the merit order curve is not
constant, hence not easy to identify.
Once the merit order curve for a certain point of time is determined the bids
21
Figure 3.3.: Two examples of the price formation via merit order curve on
the EEX on Tuesday 2012/07/17. The bids (green line) and
offers (blue line) cut at [−100, 250] cover the total price range
[−3000, 3000].
of the market participants are considered. The spot price is determined as the
price where bids and offers match (compare Figure 3.3). This price is published
by the exchange. As explained in Section 2.1 a market for forward contracts
with physical delivery exists. These contracts reduce the demand on the spot
market as well as the supply. Therefore, the price level is not affected by the
trading volumes in the forward market. The spot price is still set by the costs
of producing an additional unit of electricity.
Influence factors on the merit order curve
Apart from changes due to fuel price movements there are changes in the merit
order curve due to renewable energy. Electricity from wind, PV, biomass,
run-of-river power plants and seasonal storages is (almost) free of variable
costs. For regulatory reasons these energy sources have higher priority than
conventional energy sources. Renewable energy moves the rest of the merit
order curve to the right and takes the place of uranium. Though the installed
22
capacity is known the generation of renewable energy is unknown. Therefore,
we cannot determine how far the merit order curve is moved to the right.
Another type of power plants contributes to the variability of the merit order
curve: pumped storage power stations. As long as pumped storages are not
used like seasonal storages the assumption of zero costs for pumped storages
is not appropriate. At least the costs of pumping the water up again must be
covered by the generated electricity. As the use of pumped storages is highly
dependent on the load and the prices for pumping we cannot assign a certain
place within the merit order to pumped storages.
Due to market coupling with adjacent countries the spot price on the German
market depends on exports and imports from abroad. Currently Germany,
France, Netherlands, Belgium, Austria and Luxembourg have a simultaneous
day-ahead auction. This leads to the optimal use of border capacities as the
combined spot price auction on all markets results in trade of surplus generation capacities between these countries. With increasing imports and exports
the differences between electricity prices in these countries decrease. As a consequence not even the knowledge about the variable costs of all German power
plants would be sufficient to describe the merit order curve and the resulting
spot price exactly. Historical import and export data might be available but
forecasts for the future are not possible without consideration of all markets.
Therefore, market coupling effects are neglected within this work.
Now that we understand the construction of the merit order curve we start
considerations with respect to modeling issues. From a fundamental point of
view the spot price is the solution of an optimization problem. The market
participants optimize their outcome of the auction with respect to the restrictions given by the government, their power plants and their demand. Assumed
that commodity prices, generation capacities, variable, fixed and starting costs
of all power plants are known we can determine the optimal constellation of
operating power plants by means of an optimization algorithm. The most
expensive operating power plant determines the spot price. Ignoring imports
and exports the daily optimization problem can be stated as
min
qit
24 X
M
X
qit · f (Pt , t, i)
t=1 i=1
with respect to
M
X
qit = Dt ,
t = 1, ..., 24
i=1
where qit denotes the generation of power plant i at time t, f (Pt , t, i) describes
the costs of power plant i at time t depending on the fuel price Pt , M is the
23
number of power plants and Dt the demand at time t. Solving this problem
results in the quantities qit that need to be generated by the power plants.
Then, the spot price is determined as
St = max f (Pt , t, i) ,
{i:qit >0}
t = 1, . . . , 24.
Apart from the lack of knowledge about the costs of power plants this modeling approach is not tractable for computational reasons. For the future,
commodity prices need to be simulated in this situation. Hence, the optimization algorithm needs to run once for every point in time per path. Typical
applications of a simulation model require at least 1000 scenarios for one year.
In case of electricity this means 1000 · 8760 optimizations, i.e. this approach
takes too much computation time. Burger et al. (2007) give a more detailed
overview of optimization approaches.
As the first idea of exact determination of the merit order curve by the optimization approach is not tractable we need to focus on approximations of
the merit order curve. Either we neglect some fundamental influence factors
without loosing to much precision or we go for a mathematical approximation
which neglects fundamental influence factors. Any model needs to reproduce
the characteristics of electricity prices. Hence, we need to know about these.
Afterwards we can determine which approach might be able to incorporate the
characteristics.
Stylized facts of electricity spot prices
A closer look on the behavior of spot prices in the past reveals some stylized
facts. Some of these can be explained by the structure of demand.
• The (average) demand for electricity is higher by day than by night.
This leads to a daily seasonality of spot prices. Before the increase of
renewable energy sources the maximum demand and price was around
noon. Due to the increase of PV this price maximum disappeared and
the highest prices can be identified around 8 a.m. and 6 p.m (compare
Figure 2.3).
• Many industrial companies do not operate on weekends. Companies with
shift operation might even have less demand on Friday afternoon as the
last shift might stop early and the machines are shut down. The corresponding situation is found on Monday morning. As industrial companies
represent a major part of the total demand in Germany the described
24
weekly seasonality is reflected in the total load and with that in the
price.
• In many countries like the USA the electricity demand in summer is
higher than in winter due to air conditioning. As air conditioning is
not that popular in Germany the load differences between summer and
winter are due to the use of electricity for heating systems and light.
Hence, the load in winter is higher than in summer. The prices reflect
this yearly seasonality (compare Figure 2.2).
• The daily seasonality causes price fluctuations on an intra-day basis.
These fluctuations typically remain in a certain range. For example,
maximum prices are usually below 100 EUR/MWh. As soon as power
plants are unavailable, due to either planned maintenance or outage,
unexpected high load can cause extreme prices, so-called price spikes.
An extremely high load might be caused by extremely low temperatures.
Even a normal load level can cause spikes if the electricity generation by
wind and PV is close to zero. The reason for the high spot prices can
be found in the merit order curve. A high load, low renewable electricity
generation and unavailability of power plants lead to the right end of the
curve where it has a steep slope.
• Spot prices of electricity exhibit mean reversion. As soon as the price
increases over a longer period new power plants will be constructed,
transmission capacities to adjacent countries are increased and load will
decrease as companies try to save energy to reduce costs. These measures
lead the price back to its mean. The other way around, in case of low
spot prices exports will increase, less efficient power plants are shut down
and the load increases. Expected long term load changes are captured
by new power plants. Hence, the spot price will always fluctuate around
its mean.
• The occurrence of negative prices is the most recent development in
the German spot market of electricity. Due to the increase of renewable
energy supply exceeds the demand from time to time. A result of this
mismatch are negative prices, i.e. one gets paid to take some of the
surplus electricity. Common times for this situation are windy public
holidays or windy nights on weekends.
These characteristics need to be included by an adequate spot price model. In
the following section we give an overview of the literature on spot price models
for electricity.
25
3.2. Literature on electricity price models
As there is a wide range of literature on price models for electricity we give an
overview of some models. This allows for a classification of the model described
in Section 3.3.
Electricity prices in various markets have a number of attributes in common:
Seasonalities, mean reversion and price spikes (compare Section 3.1). These
stylized facts are described by Pilipovic (1998), Eydeland and Wolyniec (2003),
Weron (2006) and Burger et al. (2007). These authors give comprehensive
overviews concerning risk management and pricing issues on energy markets.
Schwartz (1997) and Baker et al. (1998) focus on the mean reversion of commodity prices.
Early well-known price models were published by Schwartz and Smith (2000)
and Lucia and Schwartz (2002). Lucia and Schwartz (2002) introduce one and
two factor models for (log) prices in the Scandinavian power market. These factors are responsible for short and long term variations of spot prices. Schwartz
and Smith (2000) follow a similar approach: In their model a Geometric Brownian motion determines the level of price and a mean reverting process covers
short term deviations from this level.
The seasonal behavior of electricity prices is the result of seasonal variations
of electricity demand. Therefore, various models describing the spot price as
a functional of the load are proposed. Gubina et al. (2000) describe the price
as an exponential function of the load. A more sophisticated early approach is
presented by Barlow (2002). This diffusion model is based on a supply function
derived by means of a Box-Cox-transformation. Kanamura and Ohashi (2007)
present a structural model to approximate the merit order curve. The idea of
Burger et al. (2004) is the starting point for the model in this work (compare
Section 3.3). They describe the log spot price by means of cubic splines of the
grid load adjusted by average availability of power plants. These approaches
do not consider the influence of varying commodity prices. These factors are
included by Coulon and Howison (2009). They give an analytic expression
for the bid stack function considering fuel prices, demand and capacity in two
US markets. A different approach is chosen by Emery and Liu (2002): They
analyze the cointegration between electricity and natural gas futures prices.
A new spot price model is introduced by Andersson et al. (2013). They introduce a fuel-adjusted heat-rate model. This model explains the price behavior
in normal load situations by means of the coal price. The typical price behavior in times of high load (strong increase of the price) and in times of low
26
3.2. Literature on electricity price models
load (possibly negative prices) is covered by a dual exponential function. This
function is similar to the hyperbolic sine function but provides more parameters to achieve a better fit. Fundamental factors besides load and coal prices
are neglected in this approach.
Beside this class of fundamental models there are various jump-diffusion models. The deterministic functions covering the seasonalities within these models
range from a constant term up to a complex function including daily, weekly
and yearly seasonalities. These models have some more common features: A
stochastic process covering the usual price behavior and a jump diffusion component capturing price spikes. Deng (1999), Escribano et al. (2002), Cartea
and Figueroa (2005) and Geman and Roncoroni (2006) show well-known examples of this class of models. A combination of jump diffusion and fundamental
influences is proposed by Elliott et al. (2003). They use a Markov chain to
describe the number of operating large generators. Hirsch (2009) gives a comparison of a regime-switching model, a jump-diffusion model and a normal
inverse Gaussian process with respect to the valuation of swing options.
Another common way of modeling price spikes is using so-called regime switching models. These models explain price spikes by hidden Markov chains. In
case of Benth et al. (2010) the Markov chain represents the state of economy.
The model parameters like rate and level of mean reversion depend on the state
of the Markov chain. The regime switching model by Huisman and Mahieu
(2003) contains three regimes: One regime for "normal" price behavior, one
regime for upward spikes and another regime for price movements from spike
level down to "normal" price level. There are transition probabilities for the
transitions between all regimes. E.g. the probability to move from the normal
regime to the downward regime is zero. This ensures upward spikes.
Several authors mention the occurrence of negative prices but only a few regard them in modeling issues. Schneider (2011) proposes the use of the area
hyperbolic sine transformation instead of the typical log transformation. This
trigonometric function mirrors the behavior of the logarithm to the negative
range. Close to zero it is almost linear. Therefore, this transformation keeps
most of the characteristics of the log transformation but adds the possibility
to cover negative prices. Fichtner et al. (2012) model the log price by a regime
switching model with a base, an upper jump and a lower jump regime. In the
lower jump regime some jumps are turned negative with a given probability
and their height is determined by a mixture of a log-normal and an exponential
distribution. Another model for negative prices is proposed by Fanone et al.
(2013). They use an arithmetic model including a Lï¿ 12 vy-process for upward
and downward jumps.
27
A recent more comprehensive survey of literature on electricity price models
is given by Carmona and Coulon (2012).
3.3. The new spot price model
Each of the spot price models in the literature overview given above covers
only some of the stylized facts described in Section 3.1 but none of them
covers all characteristics. The first approaches contain stochastic factors and
seasonalities. Further developments include price spikes as well as the load as a
fundamental factor. Recently, models including negative prices are published.
Some of the models incorporate single fuels as influence factors but we need
to cover all commodity prices relevant for the market. In case of the German
market we need to consider prices of coal, oil, gas and emission allowances.
As mentioned in Section 3.1 the inclusion of commodity prices prevents the
use of the optimization approach for simulation purposes. Therefore, we approximate the merit order curve by a mathematical function that depends on
the various commodity prices and the load as input variables. This idea is
similar to the existing SMaPS model by Burger et al. (2004). In the following we give a short overview of the SMaPS model and indicate the need for
improvements.
In Burger et al. (2004) the logarithm of the hourly spot price is described
by an
(S)
estimation of the merit order curve, f , a stochastic short term process Xt
(S)
and a stochastic long term process Yt
:
log St = f
Lt
(S)
(S)
, t + Xt + Yt .
vt
(3.1)
The function f is a cubic spline fitted to historic price and load data. Burger
et al. (2004) have shown that a time dependency of the price load curve improves the fit (see Figure 3.4).
Furthermore, the load Lt is divided by
relative availability of power
the average
(S)
plants vt ∈ [0, 1] to improve the fit. Xt
was chosen as a SARIMA process
with a season of 24 hours (see Definition 3.1). The uncertainty
devel of price
(S)
opment on a long term horizon is represented by the process Yt
following
a random walk with drift. Its parameters are derived from prices of future
contracts.
28
Figure 3.4.: Fit of the cubic spline (red) to empirical price and grid load data
(blue) in different periods.
As the spot price usually follows the merit order curve the basic idea of the
SMaPS model is still valid. But there are a few shortcomings that have occurred since development of the model.
• Price spikes cannot be captured by the normal distribution of the innovations of a SARIMA process.
• A model completely calibrated on logarithms of positive prices cannot
reproduce negative prices.
• Renewable energy sources, especially wind and solar power, change price
behavior. Especially, maximum prices within a day changed from noon
to morning and late afternoon.
• Fuel prices and their relations to each other as major influence factors
on the electricity price are neglected.
• Emission allowances need to be purchased for CO2 emissions and increase
the costs of coal, gas and petrol fired power plants.
29
Some of these problems can be solved by small adaptations of the model. A
heavy tailed distribution out of the class of generalized hyperbolic distributions
(see Appendix A.3.5) can be used for the innovations of the SARIMA process.
The area hyperbolic sine transformation as proposed by Schneider (2011) can
be used to reproduce negative prices. But the incorporation of commodity
prices requires major changes. Therefore, we propose a new model.
The SMaPS model approximates the merit order curve by a function of one
fundamental factor, the load adjusted by average availability of power plants,
although the merit order curve depends on many factors as described in Section
3.1. We include further variables in our approximation. This leads to an
improvement of the approximation but is still less exact than the optimization
approach including all restrictions. Our model for the spot price process, (St ),
can be written as
(S)
St = g (lt , Gt , Ct , Ot , Et ) + Xt
(3.2)
with a load component (lt ), gas price (Gt ), coal price (Ct ), oil
price (Ot ), price
(S)
of emission allowances (Et ) and a SARIMA process Xt . The daily commodity prices in this model are assumed constant within the day to match the
hourly electricity prices. As our approximation g does not exactly behave like
the real merit order curve we describe the deviations by a stochastic process.
These residuals contain the errors of our approximation. Furthermore, market
participants that include fix costs in their offers do not behave exactly like the
merit order curve would expect it based on variable costs. Such deviations
are covered by the short term process as well. In the following we describe
both model components
more detailed. The choice of the function g and the
(S)
stochastic process Xt
are deduced.
3.3.1. The polynomial approximation
The function g in Equation (3.2) describes the relationship between different
commodity prices and the load with respect to their effect on the spot price
of electricity. It is a more complex approach to estimate the merit order curve
than the SMaPS model used. Looking at Figure 3.5 suggests a simple modeling
approach like
g (lt , Gt , Ct , Ot , Et ) = a0 + a1 lt + a2 Gt + a3 Ct + a4 Ot + a5 Et
(3.3)
The spot price seems to grow at least linear in each of the variables. Estimation
of the coefficients a0 , . . . , a5 can be done via ordinary least squares regression
and the model is set up.
30
Figure 3.5.: Dependencies between grid load (top left), residual load (top
center), commodity prices and the price of electricity. Least
squares fit of a linear function (red). Daily average spot price in
case of commodity prices.
The approach of Equation (3.3) does not represent one of the main ideas of
the merit order curve: The cheapest power plants will be used first. If the gas
price is constant and the coal price increases, the spot price will follow this
price movement. But only to a certain point. Neglecting capacity constraints
electricity will only be generated by coal power plants up to a certain coal price.
Beyond this price it is cheaper to run gas power plants. As a consequence,
further coal price increases would not have an impact on the electricity price.
In analogy to this fuel switch from coal to gas, energy provider will switch
between any other fuels as soon as the switch is profitable.
We neglected the capacities of power plants in this consideration. Taking
capacities into account increases the complexity of this fuel switch: If the
constellation of commodity prices requires a switch from coal to gas and the
load is high the gas power plants might not be able to supply enough electricity
so that some coal power plants will run as well. A total fuel switch can only
be done if the capacities of both fuels match the load. Otherwise a mixture
of both fuels is used. Obviously, the impact on the price is strongly related to
the load level.
31
So far, emission allowances were not mentioned in this consideration. All kinds
of thermal power plants emit certain quantities of CO2 per MWh. As these
quantities differ among the fuel types the effect of emission allowances on the
price of electricity depends on the type of power plants running. Only if the
total load is covered by nuclear generation and renewable energies the price of
emission allowances can be neglected. As these situations play a minor role
we need to include prices of emission allowances depending on the load level
in our model.
The previous arguments suggest the inclusion of commodity prices taking into
account the load level. This requirement is not met by the approach in Equation (3.3). A linear combination of commodity prices and load is not sufficient
as a spot price model. The description of the merit order curve and the illustration in Figure 3.3 suggest a nonlinear behavior of the curve which should
be covered by the approximating function as well.
Although the choice of any complex function might considerably increase the
goodness-of-fit to historical data this is not advisable either. The computational tractability needs to be considered. On the one hand, the function needs
to be evaluated for any point of time in the future for every realization of the
commodity price model. This requires the function to be easy to evaluate.
On the other hand, the parameters of the function need to be estimated. The
method used for parameter estimation based on historical data needs to provide reliable results within reasonable time. Not any function g : R5 → R (or
even g : R6 → R, if g depends on the time) fulfills both conditions.
By choosing an adequate function from the class of polynomials we ensure that
parameters can easily be estimated via ordinary least squares regression. The
goodness-of-fit can be increased by choosing polynomials of high degrees as this
increases the number of explanatory variables in the regression. The problem
about polynomials of high degrees becomes apparent in simulations. As we
use stochastic models for each variable a degree of n would lead to Var(X n )
where X is the distribution of the variable. Var(X n ) < ∞ might not hold.
Hence, adequate scenarios cannot be generated by the model. Therefore, we
choose from the class of polynomials of degree ≤ 2. This restriction scales the
variance problem down to the question of existing moments of order four but
it maintains the opportunity to model combined influences.
The loss of accuracy by choosing a polynomial of degree ≤ 2 cannot be neglected. But we show that the goodness-of-fit of other spot price models is
still exceeded (compare Section 6.2). Furthermore, the structure of the model
allows for all features of electricity prices to be reproduced in simulations.
32
Influence of renewables
We did not take into account renewable energy sources so far. Renewables
can be distinguished with respect to their electricity generation. Run-of-river
and biomass power plants contribute a steady, slightly seasonal generation of
electricity to the market. Generation from wind and solar power plants is
subject to considerable fluctuations. Both kinds of renewable power plants
lead to a decrease of the spot price as they are free of variable costs. As
the fluctuations of water levels are comparably low the generation by run-ofriver power plants has small fluctuations and leads to a permanent decrease
of the spot price. The same is applicable to biomass power plants. Due to
the increase of fuel costs the number of biomass (and other renewable energy
sources) power plants increases. Nevertheless, once these power plants are
built they provide a steady generation.
The installed capacities of wind and solar power plants are remarkably higher
than the installed capacities of other renewables. But due to the dependency
on weather conditions their generation might even be zero. There is no reliable
long term forecast or at least "good" estimation of power generation by wind
and solar power plants. As a consequence the impact on the spot price is not
as steady as by run-of-river power plants.
Once the power plants are built the generation by renewable energy sources is
free of variable costs. In addition to this economic advantage, renewable energy
sources are supported by most governments. This means that their generation
has highest priority. Renewable power plants are not turned off at any time.
The generation by wind and PV power plants is placed in the markets by
transmission system operators. In Germany they have to sell the energy at any
price. Thus, they offer certain quantities even at a price of -3000EUR/MWh
though paying a fixed price to producers. This ensures that the first part of
consumer demand is satisfied by renewable energy sources and the rest needs
to be satisfied by conventional power plants. As a consequence of this the
usual principle of merit order takes effect after the renewable energy sources
have supplied a part of the demand. The commodity prices set the electricity
price for the grid load less the generation by renewable energy sources.
Nuclear power plants contribute a major part of the power generation in Germany. These plants cannot easily be turned off or reduced in their generation
for financial, technical and legal reasons. Therefore, these plants run nonstop
with the exception of maintenance outages. As a consequence of the nuclear
disaster in Fukushima, Japan, in March 2011 the German government forced
the shutdown of several German nuclear power plants. This means that the
33
cheapest conventional source of energy is partly replaced by more expensive
thermal power plants. In contrast to this increasing effect on the price level
there is an increase of flexibility. The generation of thermal power plants used
instead can be adjusted to react on changing infeed from renewables. This
flexibility increases the smoothness of prices. Nuclear power plants generate
electricity at almost constant prices. The fluctuations of uranium prices are
negligible and do not contribute to major electricity price movements. Therefore, we can consider the grid load less generation from renewable and nuclear
energy sources as the load that sets the price. We define residual load as
lt = Lt − P Vt − Wt − Bt − Nt
(3.4)
where (Lt ) is the grid load, (P Vt ) is the generation by solar power plants, (Wt )
is the generation by wind, (Bt ) is the generation by biomass, run-of-river and
minor renewable energy sources and (Nt ) is the generation by nuclear power
plants. This definition ensures that the effect of renewables on the price is
considered as well as the shut down of nuclear power plants.
This approach is in line with recent literature: Thoenes (2011) refers to grid
load less wind power generation as residual load. This variable and commodity
prices are used to explain the historic spot price behavior. The model is not
specified for simulation purposes. Wagner (2012) proposes the use of residual
demand (total demand minus generation from renewables) to improve model
fit of load based electricity price models. Even the fit of a linear function of
the load to the price is improved by the use of the residual load as it can be
seen in Figure 3.5.
In the explanations above all sources of energy contributing to the generation
of electricity in Germany are mentioned except of lignite. Lignite is produced
in Germany and immediately consumed for generation of electricity as transportation of lignite is too expensive. Due to the comparably low energy content of lignite huge quantities are required. Transportation of such quantities
would exceed the profitability of lignite-fired power plants. Therefore, there is
no market for lignite. The costs of production can be regarded as constant and
the capacities are almost constant. Some new power plants are built but these
do not change the total capacity of generation by lignite too much. Therefore,
the electricity generation with nearly constant capacities and production costs
will not make a major contribution to the variation of the spot price of electricity. Lignite influences the total level of the spot price but not the hourly or
daily variation. Thus, the price effect of lignite generation is constant and can
be captured by the constant in our price model. If major capacity changes of
lignite lead to a different behavior of the electricity price the residual load as
defined above might be adapted by the lignite capacities as well. Such a step
should work in analogy to the consideration of nuclear capacities.
34
Consideration of capacities for gas or coal power plants is more complex. These
power plants operate only in certain hours. Changes of capacity will only affect
these hours. As explained in the description of the merit order curve there is
no clear indication of these hours. Therefore, we assume constant capacities
of coal and gas within our data and for the time period of simulation.
A more sophisticated approach for our model in Equation (3.2) is given by
g (lt , Gt , Ct , Ot , Et , t) = a0 + a1 1Sat (t) + a2 1Sun (t)
+a3 lt + a4 Gt 1A (t) + a5 Ct + a6 Ot + a7 Et
+a8 lt Gt + a9 lt Ct + a10 lt Ot + a11 lt Et .
Apart from statistical significance there are fundamental and heuristic reasons
for the choice of these factors:
• a0 : A constant term for the average price level.
• 1Sat and 1Sun : Indicator variables equal to one on Saturdays or Sundays.
Some power plants run on weekends to avoid starting costs although their
variable costs are higher than the (expected) spot price. The coefficients
for these indicator variables take account for this situation and decrease
the price level on weekends. The goodness-of-fit is significantly increased
by the use of two indicator variables instead of one indicator describing
the weekend.
• lt , Gt , Ct , Ot and Et : Residual load as well as commodity prices have a
linear influence on the spot price. The higher the commodity prices or
the residual load, the higher the spot price (compare Figure 3.5). These
terms neglect the non-linear relationship between the variables described
above.
• 1A : A describes the hours from 7 a.m. to midnight as the statistical
significance of the gas price is only given in these hours. Gas fired power
plants are used in times of high load only due to the high variable costs
in comparison to coal fired power plants. Therefore, there is no influence
on the spot price at night. This represents the situation in our data and
neglects a possible future fuel switch from coal to gas.
• lt Gt , lt Ct , lt Ot and lt Et : The influence of the various commodity prices is
strongly related to the level of the residual load. Therefore, we include
these combinations of residual load and commodity prices.
35
The model components are chosen symmetrically. Each commodity occurs
twice in the model: alone and in combination with the residual load. All
fuel types are included due to the portfolio of generation assets in Germany.
Nevertheless, it turns out that negligence of the oil price component has a
minor effect on the goodness-of-fit. The fluctuation around the linear fit in
Figure 3.5 suggests that there is no strong dependency on the oil price. The
statistical significance of the component might be due to the correlation to
other commodities. In the model without oil prices the other components take
over the role of oil. This reduces the complexity of the function but keeps
the goodness-of-fit measured by the coefficient of determination at a level of
R2 = 0.69.
Apart from such statistical reasons there is a fundamental justification for the
negligence of oil prices in the spot price model. As the generation by oil-fired
power plants is below two percent of the total generation in the German market
(compare Figure 3.2) it is considered a minor source of electricity. The model
approach above changes to
g (lt , Gt , Ct , Et , t) = a0 + a1 1Sat (t) + a2 1Sun (t) + a3 lt + a4 Gt 1A (t)
+a5 Ct + a6 Et + a7 lt Gt + a8 lt Ct + a9 lt Et .
(3.5)
The formulation of model (3.5) as
g (lt , Gt , Ct , Et , t) = a0 + a1 1Sat (t) + a2 1Sun (t) + a4 Gt 1A (t) + a5 Ct
+a6 Et + lt (a3 + a7 Gt + a8 Ct + a9 Et ) .
allows for another heuristic explanation of the model structure. The term
lt (a3 + a7 Gt + a8 Ct + a9 Et ) describes residual load times average costs of generation. These average costs consist of a weighted average of commodity prices
representing the mixture of generation assets responsible for electricity generation in the market. The terms a4 Gt 1A (t), a5 Ct and a6 Et cover deviations from
this average due to price movements of single commodities.
Time dependent model
The goodness-of-fit of the model (3.5) measured by R2 = 0.69 can be increased
by a simple modification. Instead of using one set of parameters for the data
set it is possible to use 24 sets of parameters. One set for each hour:
g (lt , Gt , Ct , Et , t) = a0 (t) + a1 (t)1Sat (t) + a2 (t)1Sun (t)
+a3 (t)lt + a4 (t)Gt 1A (t) + a5 (t)Ct + a6 (t)Et
+a7 (t)lt Gt + a8 (t)lt Ct + a9 (t)lt Et .
36
Technically, the data set is divided into 24 data sets and the regression is done
24 times. The errors of the regression are aggregated to one time series which
is the basis for the estimation of the stochastic process. On the one hand, the
R2 increases to 0.74. On the other hand, the total number of parameters is
increased by factor 24. As the goodness-of-fit always increases as the number of
parameters increases we need to judge whether the huge number of parameters
is justified by the increased R2 . An obvious criterion would be the adjusted
coefficient of determination
R̄2 = R2 − (1 − R2 )
p
T −p−1
with the number of parameters p and the sample size T . As in our case
T > 30000, p < 300 holds the adjustment term will not change the decision
taken according to R2 . A likelihood based information criterion considering
the number of parameters like the Bayesian information criterion (BIC) is not
applicable due to the missing likelihood function in our situation. Thus, we
try to reduce the number of parameters used in the models without further
consideration of concepts of model selection.
As a first step it turns out, that the indicator variables for Saturday and
Sunday can be aggregated to one weekend indicator variable. There is no
fundamental reasons for having two different indicator variables for Saturday
and Sunday in the initial model but the R2 significantly increases. Having
only one indicator variable is what we expect and it is supported by the 24
models. In this representation the indicator variable 1A (t) is neglected as the
parameter a3 (t) might be zero, if it is not significant in certain hours.
g (lt , Gt , Ct , Et , t) = a0 (t) + a1 (t)1W e (t)
+a2 (t)lt + a3 (t)Gt + a4 (t)Ct + a5 (t)Et
+a6 (t)lt Gt + a7 (t)lt Ct + a8 (t)lt Et
(3.6)
The fluctuations of the ten parameters in the 24 models should be considered.
If these fluctuate randomly from one model to the next, it is neither possible
to call the 24 models reliable nor the initial model. Under the assumption
that the parameters in adjacent hours have a similar level we can try to model
adjacent hours together. Instead of having 24 models a smaller number of
models implying less parameters is needed.
In Figure 3.6 the parameters of the 24 models are compared. As the coefficients
of terms including the residual load as a factor are very small we compare the
parameters multiplied by the historical mean of the corresponding variables.
Terms containing the same commodity are aggregated. The results as given in
37
Figure 3.6.: Parameters of 24 models multiplied by average price and load
levels of the history. Aggregated over commodities.
Figure 3.6 meet the expectations. In times of increased residual load the gas
terms have a higher influence (8 a.m. - 8 p.m.). When the coal terms decrease
(7 a.m.) the gas terms take over the role of the fuel that sets the spot price.
The emissions terms show a similar behavior in all models. These terms lead
to an almost constant increase of the price level. The load term is responsible
for the increased price level between 6 p.m. and 8 p.m. Altogether these terms
and the constant term lead to an appropriate average price in every hour (the
weekend indicator variable is neglected).
As the estimation results for the 24 models do not show "random" fluctuations
it is reasonable to choose a time dependent modeling approach. This approach
needs to be adapted due to the high number of parameters. For this purpose
we try to identify blocks having similar parameters. The most obvious block
is formed by the first seven hours. There are no remarkable changes in the
parameters during these hours. Especially, coal and gas terms show a mirrored
behavior with several slopes. If we take hours where the slope changes its
direction as boundaries for further blocks, this choice results in the blocks 0-7,
7-12, 12-18 and 18-24. Applying the model to these four blocks leads to an R2
of 0.71. The goodness-of-fit does not change if we take 0-6 as the first block.
This block is a commonly traded OTC contract which is another indicator for
38
the similar behavior of prices in these hours.
Figure 3.7.: Parameters of models for four blocks of six hours multiplied by
average price and load levels of the history (solid lines). Parameters of the initial model for comparison (dotted). Aggregated
over commodities.
The parameters of the four models (see Figure 3.7) show similar behavior to
the parameters of 24 models discussed above. As expected, parameters of the
initial model are in between those of the four models (and the 24 models as
well).
We presented two alternatives to the initial model (3.5). These alternatives
were discussed with respect to the goodness-of-fit and the number of parameters. A final decision about the "best" model cannot be made. The alternative
using 24 parameter sets is neglected due to the high number of parameters.
Further discussion can be found in Chapter 6. In the following we refer to the
initial model unless otherwise stated. Due to the similarity of the alternatives
and the initial model the choice of the stochastic process is not affected by the
decision between these models. Though the estimated parameters differ the
process is adequate in all three cases.
39
Residuals
Using the model components given in the function g the normal price behavior
can be described. In times of major power plant outages this function is not
able to reproduce the extreme spot price. Thus, extreme prices, so-called
outliers, are excluded from the data before parameter estimation of g. Any
data being further than four times the standard deviation away from the mean
is declared to be an outlier. This simple approach is sufficient to increase the
goodness-of-fit of the model.
After fitting g to the data without outliers
St − g (lt , Gt , Ct , Et , t)
(S)
is the basis for the estimation of Xt . This means, that outliers are due to
stochastic events in our model. The stochastic process is not only supposed
to cover outliers but any kind of abnormal price behavior as a consequence
of planned or unplanned power plant outages, imports and exports, strategic
decisions of energy utilities and approximation errors due to the choice of g.
Therefore, we fit a stochastic process (see Section 3.3.2) to cover the price
range in our simulation model.
A comparison of Equation (3.2) and Equation (3.1) reveals a missing long term
factor in the new model. We discuss that long term uncertainty of electricity
prices is a result of long term uncertainty of commodity prices (see Section
6.2). Therefore, we do not need a long term process for the electricity price
but rather include adequate terms in each commodity price model.
3.3.2. The stochastic process
The difference between (St ) and g contains all unexplained influences on the
spot price. Although there might be fundamental reasons for every deviation
from g we declare them to be random. This randomness can be covered by an
adequate stochastic process which is specified below.
Although the residual time series does not have any deterministic structure
a high autocorrelation can be identified. Within the autocorrelation function
(ACF) a periodicity can be identified. All lags being multiples of 24 have a
slightly higher autocorrelation than other lags (see Figure 3.8). As the time
series is stationary we apply a seasonal autoregressive integrated moving average (SARIMA) process as defined by Brockwell and Davis (1987,
40
Chapter 9.6). The class of SARIMA processes is an extension of the ARMA
processes described in Appendix A.1.1.
Definition 3.1 (SARIMA process). If p, d, q, P, D, Q, s are non-negative integers, then (Xt ) is a SARIMA (p, d, q) × (P, D, Q)s process with period s if the
differenced process
Yt = (1 − L)d (1 − Ls )D Xt
is a causal ARMA process
φ (L) Φ (Ls ) Yt = θ (L) Θ (Ls ) Zt ,
Zt ∼ W N 0, σ 2 ,
(3.7)
where
φ(z)
Φ(z)
θ(z)
Θ(z)
=
=
=
=
1 − φ1 z − . . . − φp z p ,
1 − Φ1 z − . . . − ΦP z P ,
1 + θ1 z + . . . + θq z q ,
1 + Θ1 z + . . . + Θ Q z Q ,
z ∈ C.
Applying a SARIMA process to a time series means determination of the
model orders p, d, q, P, D, Q, s and estimation of p + q + P + Q coefficients of
the polynomials φ, θ, Φ, Θ. According to the model choice in case of ARMA
processes the orders of a SARIMA process can be determined by an information
criterion. But due to computational complexity one is interested in very small
numbers of parameters. Therefore, we try to find adequate orders by analyzing
ACF and partial autocorrelation function (PACF).
First, we can apply e.g. the Augmented Dickey-Fuller-test (see Appendix
A.1.3) to state that the residuals of the spot price process are stationary.
Hence, we do not take differences of the time series: d = D = 0. Obviously,
ACF and PACF exhibit a period with lag 24. This is not only a mathematical
finding but it can be supported by a heuristic argument: A model error in
one hour is correlated to the model error in the same hour on the day before.
Therefore, we set s = 24.
The first s lags of ACF and PACF can be used to choose p and q as it is done
in the model choice for ARMA models. Dividing the whole time series into 24
individual time series, one for each hour of the day, and analyzing 24 ACFs and
PACFs gives an idea about adequate choices for P and Q. This gives us four
orders which are separately determined. To make sure that the result is an
adequate model for the time series we check ACF and PACF of the innovations.
The innovations should be white noise without any autocorrelations (see Figure
3.8).
41
Figure 3.8.: Autocorrelation function (left) and partial autocorrelation function (right) of price residuals (blue) and innovations of the
SARIMA process (green).
According to this analysis a SARIMA (1, 0, 0)×(1, 0, 1)24 process is appropriate
to describe the features of the residual time series. The innovations of this
process, (Zt ), are uncorrelated but not normally distributed. The histogram
of the empirical data of (Zt ) (see Figure 3.9) reveals the typical shape of
a Student’s t-distribution (compare Appendix A.3.1). A result of Hannan
(1973) allows to choose different distributions than the normal distribution
for the innovations of an ARMA process as long as mean and variance of the
distribution exist (see Appendix A.1.2).
A Student’s t-distribution with scale parameter can be fitted to the data via
MLE. The variance of a Student’s t-distribution for any degree of freedom
ν > 2 is given as ν/(ν − 2). As we obtain ν = 2.7 the conditions for the
application of this distribution to the innovations of the SARIMA process are
met.
Our spot price model consisting of a polynomial with several input variables
and a stochastic short term process is specified. In the following we present
models for the input variables. We start with two modeling approaches for the
residual load.
42
3.4. Modeling the load
Figure 3.9.: Fit of a Student’s t-distribution (red line) to the histogram of
the empirical innovations of the SARIMA process of the spot
price (blue).
The spot price models presented in the previous chapter have the residual load
(see Figure 3.10)
lt = Lt − P Vt − Wt − Bt − Nt
as an input variable. In this chapter we introduce two stochastic modeling
approaches.
1. Distinct models for the grid load Lt , solar generation P Vt , wind generation Wt , miscellaneous generation Bt and nuclear generation Nt are used.
The residual load scenarios are composed of simulations of these distinct
models according to the formula above.
2. One model for the residual load lt is set up. It includes variations from
all sources of uncertainty included in the distinct variables.
From a theoretical point of view both model structures are thinkable. Therefore, we will present both approaches in the following. A discussion including
43
practical reasons as well as mathematical arguments reveals advantages and
disadvantages of both approaches. We will give this discussion partly within
this chapter. The remaining part follows in Chapter 6.
Figure 3.10.: German grid load (left) and residual load (right) in 2010 (top)
and one week in April 2012 (bottom).
The data basis for both approaches is given by grid load data provided by
European Network of Transmission System Operators for Electricity (2013).
The historical generation of renewables is given by the Transparency Platform
of the EEX (see European Energy Exchange (2013)).
3.4.1. Modeling the grid load and renewable energy
generation
The first approach requires models for all five components of the residual load.
We refrain from a model for the nuclear generation. For reasons of flexibility
the future generation by nuclear power plants is considered constant. This
approach neglects maintenance outages and outages due to technical problems.
This error is as small as the benefit from a more detailed generation forecast
for the nuclear generation. Nevertheless, the model structure gives the chance
44
for later inclusion of a different forecast. The remaining four variables are
modeled in the following.
Modeling the grid load
Modeling the load is rather about identifying adequate functions to cover various seasonalities than about finding a complex stochastic process. The grid
load exhibits the seasonalities explained in Section 3.1. Forecasting the seasonal behavior of the grid load by means of deterministic functions is easy
in comparison to other time series requiring complex model structures and
exogenous variables.
The load of a large grid is the sum of a large number of consumer loads.
To a certain extent, fluctuations of single consumers cancel each other out so
that the total is quite stable and predictable. Nevertheless, deviations from
bt , occur due to weather influences or unpredicted events. This
the forecast, L
means that weather dependent variables might even improve the model fit.
The weather dependent
deviations and forecast errors are included in a short
(L)
term process Xt . In the long run load changes due to fundamental changes
in the economy, as seen in the crisis 2008/2009, can be included by a long term
process. Altogether, we model the grid load as
bt + Xt(L) + Yt(L) .
Lt = L
(L)
Yt
(3.8)
The long term process
is supposed to cover load changes for economic
reasons. This kind of changes is characterized by long lasting changes of the
load level. An increase of load during one week is not considered as a fundamental change of economy but rather changes lasting a few months or years.
Therefore, we do not use hourly data to identify the long term fluctuations.
Long term load data with a monthly granularity is available with a longer
history than hourly data. Using this data, fluctuations due to economy can be
identified.
In order to ensure that short term and long term volatility do not influence
each other we aggregate monthly data to quarterly data. This data still contains yearly seasonality and a slight long term trend. After removal of these
statistically significant deterministic components we obtain a residual time series containing all economic influences (and potential errors of the removal
of seasonalities and trend). These residuals are stationary as we expect it in
advance: There are economic fluctuations but after a crisis economy usually
45
tends to revert to its mean. These fluctuations around the mean can be modeled by an AR(1) process (see Figure 3.11). Although we remove a linear trend
from the historical data we ignore it in our simulation. There is no clear trend
observable after 2005. Therefore, we consider the linear growth as negligible
for the future.
Figure 3.11.: German quarterly load data from January 1991 to September
2012 (top left); Quarterly load data without trend and season
(top right); Historical data and 10 simulated paths for 3 years
(bottom).
We use the long term process to model the level of load. The structure on
top of this level is given by seasonalities. The easiest way to cover all daily
and weekly seasonalities is to use a system of similar days. Each similar day is
represented by an indicator variable. Working days, Saturdays, Sundays and
public holidays are the most popular choices for similar days. One has to decide
whether all working days are represented by one indicator variable or by up to
five indicator variables. Such a system covers the weekly seasonality. As we
work on hourly data each similar day is not only represented by one indicator
variable but rather by 24 to specify all 24 hours of the similar day. If differences
between the load structure in different periods of the year exist, one set of
similar day for each period can be chosen. Further possibilities to incorporate
the yearly seasonality are the use of 12 indicator variables on monthly basis or
46
a combination of sine and cosine to have a continuous function.
Depending on the choice of similar days, the number of periods within a year
and the type of function for the yearly seasonality the system of variables
bt by linear regression a big system
might be quite large. As we want to fit L
of regressor variables can lead to computational problems (close to singular
matrices). As the load is comparatively easy to predict even a rather small
system of regressors captures the most relevant characteristics. We use sine
and cosine for the yearly seasonality, similar days for the weekly seasonality
and 24 indicator variables per similar day for the daily seasonality. For the
comparison of different models we choose the R2 as a measure of goodness-offit. Holidays, single days between holidays and weekends, working days and
weekends in summer, winter and spring/autumn give us a system of similar
days leading to R2 = 0.89. The same system applied to the grid load model
without a long term process results in R2 = 0.92. Seven similar days in four
periods already produce a rather high goodness-of-fit. Further improvements
can be achieved by more similar days or periods and the inclusion of weather
dependent exogenous variables. The model improvement by inclusion of further similar days decreases. For our purposes this simple modeling approach
is sufficient. A detailed description of load modeling techniques with further
references is given by Weron (2006). Alfares and Nazeeruddin (2002) provide
a survey and classification of load models.
As it can be seen above the inclusion of the long term component reduces the
goodness-of-fit in the period of time considered in this work. As the component
has a fundamental justification we keep it in the model. The representation
of long term fluctuations is an essential part of the grid load model. According to the state of the economy the demand for electricity fluctuates. Such
fluctuations on a long term horizon need to be covered by a grid load model.
The presentation within this chapter slightly differs from Chapter 7. In Chapter 7 the long term component is based on monthly data instead of quarterly
data. Nevertheless, the purpose behind both ways is the same. Especially,
during the crisis 2008 the economic load fluctuations could be explained by
the long term component and the model fit was improved. The use of the
component decreases as the data used in this work starts in 2009. But still,
load fluctuations have a major contribution to the level of the spot prices and
the riskiness of retail power contracts.
bt leads to the
Estimation of long term process and deterministic function L
residual time series. This part of the model contains all unpredictable fluctuations due to short term load changes (see Figure 3.12).
47
Figure 3.12.: German grid load and forecast from 2012/3/12 till 2012/3/18
(top left) and from 2012/8/27 till 2012/9/2 (top right). The
empirical load short term process (bottom).
The short term effects usually do not occur in single hours but in blocks of
hours. Therefore, we need to analyze the autocorrelations within the time
series. This reveals strong autocorrelations. On the one hand, lags of 1-3
hours are significant. On the other hand, lags of (multiples of) 24 hours show
an increased autocorrelation (see Figure 3.13). The former can be explained
by the fact that events leading to short term load changes usually last more
than one hour and influence a few adjacent hours as well. The latter is a result
of the model specification: The daily structure of load changes within a year
e.g. due to the decreasing need for electric lighting in summer. We include
various periods for our similar day system to cover these changes but they are
somehow continuous and not stepwise as it is suggested by the similar day
system. Therefore, a model error in any hour on one day is likely to occur in
the same hour on the days before as well.
As the ADF-test indicates stationarity of the residual time series we choose
a seasonal ARMA process with period 24. The model order of this process
is determined as described above in case of the residual price time series (see
Section 3.3.2). The result is a SARIMA (2, 0, 0) × (1, 0, 1)24 process. After
determination of the model order an adequate distribution for the innovations
48
Figure 3.13.: ACF (left) and PACF (right) of residual time series (blue) and
innovations of SARIMA process (green) for the grid load.
of the process has to be fitted. We use a Student’s t-distribution with scale
parameter to account for the increased variance in comparison to the standard
t-distribution. The parameter of the t-distribution (degrees of freedom) allows
for a good fit to the empirical distribution, even in the tails (see Figure 3.14).
The grid load is not likely to show any extreme values. Extreme values occurring in the residual time series or in the innovations of the process are due to
forecast errors. The original load data does not exhibit any extreme values or
outliers (see Figure 3.10). Therefore, a truncated version of the t-distribution
leads to more realistic simulation results. We choose the range of the empirical
distribution as the support of the t-distribution.
Altogether these model components form a reliable model to reproduce the
characteristics of the grid load (see Figure 3.15).
Modeling wind and PV generation
The generation of wind and PV power plants is characterized by strong weather
dependent fluctuations. Reliable forecast models of weather conditions are
49
Figure 3.14.: Fit of a truncated Student’s t-distribution (red line) to the
histogram of the empirical innovations of the SARIMA process
of the load (blue). The figure is restricted to [−2000, 2000].
available for a few days only. Hence, a simulation model for a few years cannot
easily be constructed by a forecast plus a stochastic process as it is done in
case of the grid load (see Section 3.4.1).
The major drivers for generation of wind and PV power plants are weather
situation and installed capacities. As the installation of renewable capacities
is supported by most governments there is a strong increase that is not expected to stop within the next few years. Thus, it is not easy to identify a
stochastic process describing weather dependent generation based on given installed capacities. Therefore, we use a pragmatical method to obtain realistic
simulations.
The basic concept of our model is historical simulation. Historical data contains realistic fluctuations which is an essential condition for any model. Nevertheless, there are two problems concerning the use of historical data.
1. There is only a short history of PV and wind generation data (less than
five years, compare Figure 3.16).
50
Figure 3.15.: Two scenarios of the grid load for 2013.
2. Historical capacities are not valid for the future.
In order to extend the pool of data that we can draw from we generate further
data. Weather data from the last 20 years is used to derive wind and PV generation data under the assumption of current capacities. With this extension
a historical simulation generates sufficient variations.
Though future capacities are not exactly known there are forecasts. Thus,
we can scale the historical scenarios to expected future capacities. As the
forecast is subject to uncertainty we include fluctuations of the forecast in our
model. In combination with the historical simulation this ensures sufficient
stochastic behavior of the resulting scenarios which are subtracted from grid
load scenarios.
More sophisticated models can be applied instead of this simple approach
as soon as a longer history of data and a consistent generation capacity are
available.
51
Figure 3.16.: Historical wind (top) and PV (bottom) generation.
Modeling miscellaneous renewable generation
Miscellaneous renewable generation describes sources such as run-of-river and
biomass power plants. Further sources of energy with minor quantities are
included as well. The increase of capacities is small due to the availability
of the corresponding fuels. Most profitable locations for run-of-river power
plants are already used. This reduces the speed of extension. Therefore, it is
acceptable to assume the capacities as constant for the next few years.
Neglecting the increase of biomass power plants we can assume constant capacities of miscellaneous renewable generation. Apart from seasonal variations by
run-of-river power plants there is almost no fluctuation in miscellaneous generation. Thus, we assume the generation of the last year in the historical data
to be continued for the years to be simulated. This is analog to the approach
for nuclear capacities.
Using the historical simulation models for renewable generation and the model
for the grid load allows for calculation of the residual load. These scenarios
contain estimations of the increase of renewable capacities as well as load
fluctuations due to various short term influences (see Figure 3.17).
52
Figure 3.17.: Two residual load scenarios for 2013. The residual load is composed by scenarios from the grid load model and the models
for renewables.
The model for miscellaneous renewable generation is the last part needed for
the first modeling approach for the residual load. Distinct models for each
residual load component were presented. In the following the second approach
with only one model is introduced.
3.4.2. Modeling the residual load
The residual load as defined in Equation (3.4) contains the grid load as a major
variable. Therefore, grid load and (lt ) contain the same seasonalities. In case
of residual load, these seasonalities are "disturbed" by renewable generation.
If the grid load model is applied to the residual load the goodness-of-fit of the
similar day system described in Section 3.4.1 decreases. This is due to the
disturbances by renewables. They reduce the predictability.
The mismatch of the grid load model results in increased errors. Hence, the
variance of the stochastic process increases. But this is what we expect: Solar
and wind power are unpredictable and lead to an increased volatility in the
53
model. Thus, the stochastic short term process in the model for (lt ) contains
grid load volatility as well as solar and wind power generation volatility.
These considerations justify the use of the same similar day system as in case
of the grid load to obtain the forecast b
lt in
(l)
lt = b
lt + Xt .
(3.9)
Thus, we use the similar day system and a SARIMA-process
(l)
Xt
to describe
2
the residual load. The goodness-of-fit measured by R reduces to 0.72 in
comparison to 0.92 for the grid load.
The scenarios resulting from the grid load model applied to the residual load
can be seen in Figure 3.18. Due to the structure of the model, the increase
of renewable capacities is not reflected in the model. The obvious loss of the
Figure 3.18.: Two realizations of the grid load model applied to the residual
load for 2013.
clear yearly seasonality can be explained by the decreased model fit. The
lower load level in summer is not identified by the model due to the renewable
energy sources. Therefore, the scenarios cannot reproduce this feature of the
historical input data.
54
3.5. Inclusion of futures market information
The missing increase of renewable capacities is the major drawback of this
model. The simplicity of having only one model instead of several models as
in Section 3.4.1 cannot compensate this drawback. Another drawback occurs
in Chapter 7.
In the following we present a method to include current information from the
futures market in our price models.
3.5. Inclusion of futures market information
An adequate price model is supposed to cover the (stochastic and deterministic) price behavior of the commodity in the past as well as the expected price
level for the future. As models are usually set up on historical prices some kind
of average of those prices or the latest available price is used as the expected
future price level. This expectation from the past might be a good estimator
of the "true" price level, i.e. the level that will realize.
Any expectation resulting from a model can only contain the information that
is in the model. Therefore, information about future changes in the market
structure, political interventions and other fundamental changes might not be
included in this expectation. All information available to market participants
is included in the prices of future and forward contracts. These prices can be
seen as the market’s expectation about the future price level.
As prices of future contracts are available for all considered commodities and
electricity we introduce the incorporation of these prices in a general way before
the individual models are described. After estimation of all model parameters
sets of scenarios can be generated from a model. The mean of these scenarios
in every hour (on every day) is shifted to the price of the future contract for
the corresponding period of time, i.e. we need to determine bt in
n
1 X (i)
x + bt = F (t) ∀ t
n i=1 t
(3.10)
(i)
with xt the i-th realization of a random variable Xt , n the number of scenarios
and F (t) the price of future contracts for time t. F (t) is either a stepwise
function consisting of the different prices of future contracts for different times
of delivery or a continuous function containing the expected price structure
within each delivery period. In the latter case the average price in each delivery
period corresponds to the price of the price of the future contract for this
55
delivery period (compare Section 7.2.1). By setting
(i)
(i)
x
et = xt + bt
i = 1, . . . , n, ∀ t
the futures market information is incorporated into the scenarios from our
model.
To be consistent with the model structure we only apply the shift to additive
models. In case of multiplicative models, i.e. models for log prices, the summand bt in Equation (3.10) turns into a scaling factor. In the following all
scenarios from the proposed models are either scaled or shifted to the prices
of the corresponding future contracts until further notice.
56
4. Modeling the spot price of
natural gas
For inclusion in the electricity price model a spot price model for gas is needed.
In the following we present such a model including temperature and oil price as
exogenous factors. The chapter is based on the work of Hirsch et al. (2013).
4.1. Introduction
During the last years trading of natural gas has become more important. The
traded quantities OTC and on energy exchanges have strongly increased and
new products have been developed. For example, swing options increase the
flexibility of suppliers and they are used as an instrument for risk management purposes. Important facilities for the security of supply are gas storages.
The storages are filled in times of low prices and emptied in times of high
prices which usually coincides with low and high consumption of natural gas.
Although the liquidity is comparably low, an OTC market for gas storages
exists. This means, physical or virtual gas storage contracts are traded.
These two examples of complex options illustrate the need of reliable pricing
methods. Both options rely on non-trivial trading strategies where exercise
decisions are taken under uncertainty. Therefore, analytic pricing formulas
cannot be expected. The identification of an optimal trading strategy under
uncertainty is a typical problem of stochastic dynamic programming, where
even numerical solutions are difficult to obtain due to the curse of dimensionality. Therefore, simulation based approximation algorithms have been
successfully applied in this area. Longstaff and Schwartz (2001) introduced
the least square Monte Carlo method for the valuation of American options.
Meinshausen and Hambly (2004) extended the idea to Swing options, and
Boogert and de Jong (2008) applied it to the valuation of gas storages. Their
least-squares Monte Carlo algorithm requires a stochastic price model for daily
spot prices generating adequate gas price scenarios. We prefer this approach
57
to methods using scenario trees or finite differences as it is independent of the
underlying price process.
The literature on stochastic gas price models is dominated by purely stochastic
approaches. The one and two factor models by Schwartz (1997) and Schwartz
and Smith (2000) are general approaches applicable to many commodities, such
as oil and gas. Cortazar and Schwartz (2003) present a three factor model for
the term structure of oil prices. These models can be applied to gas prices
as well. The various factors represent short and long term influences on the
price.
Extensions of these factor models are given by Jaillet et al. (2004) and Xu
(2004). Especially the inclusion of deterministic functions to cover seasonalities within gas prices is considered. Cartea and Williams (2008) introduce a
two factor model including a function for the seasonality. Their focus is on the
market price of risk. An important application of gas price models is the valuation of gas storage facilities. Within this context, Chen and Forsyth (2010)
and Boogert and de Jong (2011) propose gas price models. Chen and Forsyth
(2010) analyze regime-switching approaches incorporating mean-reverting processes and random walks. The class of factor models is extended by Boogert
and de Jong (2011). The three factors in their model represent short and long
term fluctuations as well as the behavior of the winter-summer spread.
In contrast to these models Stoll and Wiebauer (2010) propose a fundamental
model with temperature as an exogenous factor. They use the temperature
component as an approximation of the filling level of gas storages which have
a remarkable influence on the price. In this chapter we will extend the model
of Stoll and Wiebauer (2010) by introducing another exogenous factor to their
model: the oil price. There are at least two reasons why we believe this to
be useful. The main reason is that an oil price component can be considered
as a proxy that covers the dependence of the gas price on the state of the
world economy in the near future. In contrast to other indicators such as the
gross domestic product the oil price is available on a daily basis at least for
the 36 front months. Furthermore, the prices for gas imports in countries such
as Germany are oil price indexed. This fundamental link between gas and oil
prices is covered by our oil component as well.
The rest of the chapter is organized as follows. In Section 4.2 we present
the model by Stoll and Wiebauer (2010) including a short description of their
model for the temperature component. In their model the influence of the
temperature is described by so-called normalized cumulated heating degree
days. In Section 4.3 we describe our new oil price component and discuss the
need for an oil price component in the model. The choice of the component
58
is explained as well. After introducing the stochastic processes that we use
to model oil prices and temperature, we finally fit the model to data from
the TTF market in Section 4.4. It turns out that for the innovations of the
process a heavy tailed distribution like the normal inverse Gaussian (NIG) is
more appropriate than the classical normal distribution. We finish with a short
conclusion in Section 4.5.
4.2. A review of the model by Stoll and
Wiebauer (2010)
Modeling the price of natural gas in Central Europe requires knowledge about
the structure of supply and demand. On the supply side there are only a
few sources in Central Europe while most of the natural gas is imported from
Norway and Russia (compare Section 2.3). On the demand side there are
mainly three groups of gas consumers: Households, industrial companies and
gas fired power plants. While households only use gas for heating purposes
at low temperatures, industrial companies use gas as heating and process gas.
Households and industrial companies are responsible for the major part of total
gas demand.
These two groups of consumers cause seasonalities in the gas price:
• Weekly seasonality: Many industrial companies do not need gas on weekends. Their operation is restricted to working days.
• Yearly seasonality: Heating gas is needed in winter when temperatures
are low.
An adequate gas price model has to incorporate these seasonalities as well as
stochastic deviations of these.
Stoll and Wiebauer (2010) propose a model meeting these requirements and
incorporating another major influence factor: the temperature. To a certain
extent the temperature dependency is already covered by the deterministic
yearly seasonality. This component describes the direct influence of temperature: The lower the temperature, the higher the price. But the temperature
influence is more complex than this. A day with average temperature of zero
degrees Celsius at the end of a long cold winter has a different impact on the
price than a daily average of zero at the end of a "warm" winter. Similarly, a
59
cold day at the end of a winter has a different impact on the price than a cold
day at the beginning of the winter.
The different impacts are due to gas storages which are essential to cover the
demand in winter. The total demand for gas is higher than the capacities of
the gas pipelines from Norway and Russia. Therefore, gas provider use gas
storages. These storages are filled during summer (at low prices) and emptied
in winter months. At the end of a long and cold winter most gas storages will
be rather empty. Therefore, additional cold days will lead to comparatively
higher prices than in a normal winter.
The filling level of all gas storages in the market would be the adequate variable
to model the gas price. However, these data are not available as they are
private information. Therefore, we need a proxy variable for it. As the filling
levels of gas storages are strongly related to the demand for gas which in turn
depends on the temperature, an adequate variable can be derived from the
temperature.
Stoll and Wiebauer (2010) use normalized cumulated heating degree
days to cover the influence of temperature on the gas price. They define
a temperature of 15 ◦ C as the limit of heating. Any temperature below 15 ◦ C
makes households and companies switch on their heating systems. Heating
degree days are measured by HDDt = max (15 − Tt , 0) where Tt is the average
temperature of day t. As mentioned above the impact on the price depends
on the number of cold days observed so far in the winter. In this context we
refer to winter as the 1st of October and the 181 following days till end of
March. We will write HDDd,w for HDDt , if t is day number d of winter w.
Cumulation of heating degree days over a winter leads to a number indicating
how cold the winter was so far. Then we can define the cumulated heating
degree days on the day d in winter w as
CHDDd,w =
d
X
HDDk,w for 1 ≤ d ≤ 182.
k=1
The impact of cumulated heating degree days on the price depends on the
comparison to a normal winter. This information is included in normalized
cumulated heating degree days
w−1
Λd,w
1 X
CHDDd,` for 1 ≤ d ≤ 182.
= CHDDd,w −
w − 1 `=1
We will use Λt instead of Λd,w for simplicity, if t is a day in a winter. The
definition of Λt for a summer day is described by a linear return to zero during
60
summer. This reflects the fact that we use Λt as a proxy variable for filling
levels of gas storages. Assuming a constant filling rate during summer we thus
get the linear part of normalized cumulated heating degree days (see Figure
4.1). Positive values of Λt describe winters colder than the average. (Λt ) is
Figure 4.1.: Normalized cumulated heating degree days in Eindhoven,
Netherlands, for 2003-2012.
included into the gas price model by a regression approach. As the seasonal
components and the normalized cumulated heating degree days are linear with
respect to the parameters we can use ordinary least squares regression for
parameter estimation. The complete model can be written as
bt + α · Λt + Xt(G) + Yt(G)
Gt = G
(4.1)
bt , the norwith the day-ahead price of gas Gt , the deterministic seasonality G
(G)
malized cumulated heating degree days Λt , an ARMA process Xt and a
(G)
Geometric Brownian motion Yt . For model calibration day-ahead gas prices
from TTF and daily average temperatures from Eindhoven, Netherlands, are
used. The fit to historical prices before the crisis can be seen in Figure 4.2.
Outliers have been removed (see Section 4.4 for details on treatment of outliers).
61
Figure 4.2.: The model in Equation (4.1) fitted to TTF prices from 20032009.
4.3. The oil price dependence of gas prices
The model described in Equation (4.1) is capable to cover all influences on
the gas price related to changes in temperature. But changes of the economic
situation are not covered by that model. This was clearly observable in the
economic crisis 2008/2009 (see Figure 4.5). During that crisis the demand for
gas by industrial companies in Central Europe was falling by more than 10
percent. As a consequence the gas price rapidly decreased more than 10 Euro
per MWh.
The oil price showed a very similar behavior in that period. Economic changes
are the main driver for remarkable changes of the oil price level. Short term
price movements caused by speculators or other effects cause deviations from
the price level that represents the state of the world economy. Therefore, the
gas price is not influenced by daily changes of the oil price but rather by long
term changes of the oil price level. Such an influence can be modeled by means
of a moving average of past oil prices. The averaging procedure removes short
term price movements if the averaging period is chosen sufficiently long. The
result is a time series containing only the long term trends of the oil price.
62
4.3. The oil price dependence of gas prices
Using such an oil price component in a gas price model explains the gas price
movements due to changes of the economic situation.
Another important argument for the use of this oil price component is based
on Central European gas markets. Countries such as Germany import gas
via long term oil price indexed supply contracts. The import price of gas in
these contracts usually is an average of past oil prices. The pricing in import
contracts is done via oil price formulas. These formulas consist of three
parameters:
1. The number of averaging months. The gas price is the average of past
oil prices within a certain number of months.
2. The time lag. Possibly there is a time lag between the months the average
is taken of and the months the price is valid for.
3. The number of validity months. The price is valid for a certain number
of months.
An example of a 3-1-3 formula is given in Figure 4.3.
Figure 4.3.: In a 3-1-3 formula the price is determined by the average price
of 3 months (March to May). This price is valid for July to
September. The next day of price fixing is the 1st of October.
The formulas used in the import contracts are not known to all market participants. Theoretically any choice of three natural numbers is possible. But
from other products, like oil price indexed swing options, we know that some
formulas are more popular than others. Examples of common formulas are
3-1-1, 3-1-3, 6-1-1, 6-1-3 and 6-3-3. Therefore, we assume that these formulas
are relevant for import contracts as well.
As there are many different import contracts with possibly different oil price
formulas we cannot ensure that one of the mentioned formulas is responsible
for the price behavior on the market. The mixture of different formulas might
63
affect the price in the same way as one of the common formulas or a similar
one.
Evaluation of the oil price formula leads to price jumps every time the price
is fixed. The impact on the gas price will be more smooth, however. The
new price determined on a fixing day is the result of averaging a number of
past oil prices. The closer to the fixing day the more prices for the averaging
are known. Therefore, market participants have reliable estimations of the
new import price. If the new price will be higher it is cheaper to buy gas in
advance and store it. This increases the day-ahead price prior to the fixing
day and leads to a smooth transition from the old to the new price level in the
day-ahead market.
This behavior of market participants leads to some smoothness of the price. In
order to include this fact in a model a smoothed oil price formula can be used.
A sophisticated smoothing approach for forward price curves is introduced by
Benth et al. (2007). They assume some smoothness conditions in the knots
between different price intervals. It is shown that splines of order four meet
all these requirements and make sure that the result is a smooth curve. As
oil price formulas are step functions like forward price curves this approach is
applicable to our situation.
If the number of validity months is equal to one it is possible to use a moving
average instead of a (smoothed) step function to simplify matters (see Figure
4.4). This alternative is much less complex than the approach with smoothing
by splines, and delivers comparable results. Therefore, the simpler method is
applied in case of formulas with one validity month.
In the next section we will compare various oil price formulas regarding their
ability to explain the price behavior in the gas market.
4.4. Model calibration with temperature and
oil price
After justification of the oil price component as a fundamental factor for our
model we need to choose a way to include it in our existing model. Therefore,
we compare different oil price formulas in the regression model in order to find
the one explaining the gas price best.
For the choice of the best oil price formula we use the coefficient of deter-
64
4.4. Model calibration with temperature and oil price
Figure 4.4.: The price of oil (blue), the 6-0-1 oil price formula (red) and the
moving average of 180 days (green).
mination R2 as the measure of goodness-of-fit. We choose the reasonable oil
price formula leading to the highest value of R2 . Reasonable in this context
means, that we restrict our analysis to formulas that are equal or similar to
the ones known from other oil price indexed products (compare Section 4.3).
The result of this comparison is a 6-0-1 formula (see Figure 4.6). Although
this is not a common formula there is an explanation for it: The gas price
decreased approximately six months later than the oil price in the crisis. This
major price movement needs to be covered by the oil price component. As
explained above we replace the step function by a moving average. Taking the
moving average of 180 days is a good approximation of the 6-0-1 formula. All
in all, the oil price component increases the R2 from 0.35 to 0.83 (see Figure
4.5). Even if the new model is applied to data before the crisis the oil price
component is significant. In that period the increase of R2 is smaller but still
improves the model.
These comparisons give evidence that both considerations in the previous section are valid. The included oil price component can be seen as the smoothed
version of a certain oil price formula. At the same time it can be considered
as a variable describing economic influences indicated by trends and level of
the oil price.
65
Figure 4.5.: The model of Stoll and Wiebauer (2010) (green line) and our
model (red line) fitted to historical gas prices (blue line).
Therefore, we model the gas price by the new model
(G)
b t + α 1 Λ t + α 2 Ψt + X t
Gt = G
(4.2)
with Ψt being the oil price formula.
For parameter estimation of our model we use day-ahead gas prices from TTF.
Trading on TTF has a longer history of high trading volumes than the neighboring markets. Data from 2003-2012 is used for this model. Temperature data
from Eindhoven, Netherlands, is available from 1969-2012. For the estimation
of the oil price component we use prices of Brent traded on the IntercontinentalExchange (ICE). The data is available with a longer history than gas prices.
We use data from 2002-2012. Using this data we can estimate all parameters
applying ordinary least squares regression after some outliers are removed from
the gas price data, (Gt ).
Due to e.g. technical problems or a fire at a major gas storage the gas price
deviated from its normal price level which was determined by temperature
and oil price formulas. Thus, we exclude the prices on these occasions by an
outlier treatment proposed by Weron (2006). Values outside a range around a
running median are declared to be outliers. The range is defined as three times
66
the standard deviation. The identified outliers are excluded in the regression.
We do not remove them from our model, however, as they are still included in
the estimation of the parameters of the remaining stochastic process.
Figure 4.6.: Comparison of different oil price components in the model (4.2):
6-0-1 formula (red line), 6-1-1 formula (green line) and 3-0-1
formula (black line) fitted to the historical prices (blue line).
Altogether these model components give fundamental explanations for the
historical day-ahead price behavior. Short term deviations are included by
a stochastic process (see Subsection 4.4.3). Long term uncertainty due to
the uncertain development of the oil price is included by the oil price process.
Therefore, our model is able to generate reasonable scenarios for the future (see
Figure 4.7). We will specify the stochastic models
for
the exogenous factors
(G)
(Ψt ) and (Λt ) as well as the stochastic process Xt
in the following.
4.4.1. Oil price model
Oil prices show a different behavior than gas prices. This influences the choice
of an adequate model. The most obvious fact is the absence of any seasonalities
or deterministic components. Therefore, we model the oil price without a
deterministic function or fundamental component. Another major difference
67
Figure 4.7.: Realizations of the gas price process for 2013.
affects the stochastic process. While the oil price and also logarithmic oil prices
are not stationary the gas price is stationary after removal of seasonalities and
fundamental components.
A very common model for non-stationary time series is the Brownian motion
with drift applied to logarithmic prices. Drift and volatility of this process
can be determined using historical data or by any estimation of the future
volatility. For a stationary process, the use of an Ornstein-Uhlenbeck process
or its discrete equivalent, an AR(1) process, is an appropriate simple model.
A combination of these two simple modeling approaches is given by the two
factor model by Schwartz and Smith (2000). They divide the log price into
two factors: one for short term variations and one for long term dynamics.
ψt = exp (χt + ξt )
with an AR(1) process χt and a Brownian motion ξt . These processes are
correlated. We apply this two factor model as it considers long and short term
variations. The estimation of parameters in this model is more complex. The
factors are not observable in the market. Following the paper by Schwartz and
Smith (2000) we apply the Kalman filter for parameter estimation.
68
The resulting process (ψt ) is used to derive the process (Ψt ) in Equation
(4.2).
4.4.2. Temperature model
When modeling daily average temperature we can make use of a long history
of temperature data. Here, a yearly seasonality and a linear trend can be
identified. Therefore, we use a temperature model closely related to the one
proposed by Benth and Benth (2007).
2πt
2πt
(T )
+ a4 cos
+ Xt
(4.3)
Tt = a1 + a2 t + a3 sin
365.25
365.25
(T )
with Xt
being an AR(3) process. The model fit with respect to the deterministic part (ordinary least squares regression) and the AR(3) process is
shown in Figure 4.8. The process (Tt ) (see Figure 4.9) is then used to define the
derived process (Λt ) of normalized cumulated heating degree days as described
in Section 4.2.
4.4.3. The residual stochastic process
The fit of normalized cumulated heating degree days, oil price formula and
deterministic components to the gas price via ordinary least squares regression
(see Figure 4.10) results in a residual time series. These residuals contain all
unexplained, "random" deviations from the usual price behavior.
The residuals exhibit a strong autocorrelation to the first lag. Further analysis
of the partial autocorrelation function reveal an ARMA(1,2) process providing
a good fit (see Figure 4.11). The empirical innovations of the process show
more heavy tails than a normal distribution (compare Stoll and Wiebauer
(2010)). Therefore, we apply a distribution with heavy tails. The class of generalized hyperbolic distributions including the NIG distribution was introduced
by Barndorff-Nielsen (1978). The normal-inverse Gaussian (NIG) distribution
leads to a remarkably good fit (see Figure 4.11). Recall that a random variable
X is NIG-distributed if there is a representation
√
d
X = µ + βY + Y Z
with Z ∼ N (0, 1) and Y ∼ N − (−1/2, δ 2 , α2 − β 2 ), the inverse Gaussian distribution as a special case of the generalized inverse Gaussian distribution. More
details on this class of distributions can be found in the Appendix A.3.
69
Figure 4.8.: Top: Fit of deterministic function (green line) to the historical daily average temperature (blue) in Eindhoven, Netherlands.
Bottom: Autocorrelation function (left) and partial autocorrelation function (right) of residual time series (blue) and innovations of AR(3) process (green).
Both the distribution of the innovations and the parameters of autoregressive
processes are estimated using maximum likelihood estimation.
4.5. Conclusion
The spot price model by Stoll and Wiebauer (2010) with only temperature
as an exogenous factor is not able to explain the gas price behavior during
the last years. We have shown that adding an oil price component as another
exogenous factor remarkably improves the model fit. It is not only a good
proxy for economic influences on the price but also approximates the oil price
formulas in gas import contracts in Central European gas markets. These
fundamental reasons and the improvement of model fit give justification for
the inclusion of the model component. The resulting simulation paths from
the model are reliable.
70
4.5. Conclusion
Figure 4.9.: Historical temperatures and three realizations of the temperature model.
Figure 4.10.: Fit of deterministic function and exogenous components (green
line) to the historical gas price (blue).
71
Figure 4.11.: Top: ACF (left) and PACF (right) of residual time series (blue)
and innovations of ARMA(1,2) process (green). Bottom: Fit
of NIG distribution (green) to empirical innovations (blue).
72
The simulation of electricity prices using the models introduced in Chapter
3 requires scenarios of commodity prices as input. After introduction of a
gas price model in Chapter 4 we present different modeling approaches for
the remaining commodities oil, coal and emission allowances in the following.
In combination with gas price scenarios these models are the basis for the
electricity price simulation.
The markets of coal and oil are global markets as mentioned in Chapter 2.
Therefore, these prices are influenced by the state of the world economy. An
increasing world economy leads to an increasing demand for coal and oil. This
results in increasing prices. The other way round, the prices decrease in times
of a decreasing world economy. This is the long term relationship between
both commodities. For various reasons, such as supply problems due to wars
or natural disasters in production areas concerning only one commodity, the
prices might diverge for some time. Afterwards, their common movements are
kept up.
The use of coal and oil in power plants makes the producer of power buy emission allowances. As there is a high correlation between the German economy
and the world economy an increasing demand for coal and oil worldwide is
usually reflected in Germany as well. Therefore, more coal and oil are used
for power generation and more emission allowances are needed. This makes
the price of emission allowances follow the prices of coal and oil. While the
coal and oil market are global markets the market for emission allowances is
more local. Thus, the German economy moving in a different direction than
the world economy makes the prices of emission allowances and the fuels coal
and oil diverge. Nevertheless, there are remarkable correlations between all of
these variables.
The choice of future contracts, e.g. first front month or 36th front month,
is critical for the correlation. Prices of future contracts further in the future
are usually higher correlated. These prices are less disturbed by short term
influences. Another chance to reduce short term influences is the consideration
of five-day returns instead of daily returns. Correlations of daily log returns of
73
front month contract prices are given in Table 5.1. The modeling approaches
introduced in this chapter are based on different future contracts. Hence,
the correlations in the model differ as well. A model having only short term
contracts as input parameters cannot reflect a long term correlation and vice
versa.
Oil
Oil
1
Coal 0.30
Gas 0.11
CO2 0.18
Coal
0.30
1
0.35
0.20
Gas CO2
0.11 0.18
0.35 0.20
1
0.20
0.20
1
Table 5.1.: Empirical correlations of daily log returns of the front month
prices of oil, coal, natural gas and CO2 emission allowances from
2008-2012.
The correlations of short term contracts such as the first front month are usually lower than correlations of long term contracts. The above correlations are
already significant. Therefore, a combined commodity price model is required.
Independent models for each commodity would be an appreciated simplification but ignoring these correlations increases the model error. As the oil price
resulting from a combined commodity price model is an exogenous factor for
the gas price model in Chapter 4 there are dependencies between coal, emission allowances and gas as well. Thus, we may exclude gas from the modeling
approaches as long as the gas price model in Chapter 4 is used.
As these three commodities (gas is only considered in the model in Section
5.4.3) do not exhibit any deterministic trends or seasonalities, the following
models are pure stochastic approaches. In Sections 5.2 and 5.3 two well-known
approaches are suggested. In Section 5.4 the concept of cointegration as introduced by Granger (1981) is described. This concept allows for a combined
price model of oil, coal, gas and emission allowances.
5.1. Literature on commodity price models
Combined commodity price models focus on the common features of the commodities included. These models make a compromise of covering the dependencies and exact modeling of distinct features of the commodities. The literature provides either models for a combination of certain commodities or
models for distinct commodities. We give a short overview of models for some
commodity combinations and specific models for distinct commodities. To our
74
5.2. Three-dimensional random walk
knowledge there is no publication on the combination of commodities needed
for the model in this work.
The stylized facts of oil and coal prices are similar. Seasonalities do not play
a role on both markets, the most relevant fundamental driver is the world
economy. Therefore, the same models can be applied to both commodities.
The possibly higher volatility of oil prices for being subject to speculations
can be covered by a higher model volatility. Commonly used models are the
ones proposed by Schwartz (1997) and Schwartz and Smith (2000). These
stochastic models have one or two stochastic factors to model the price features.
Cortazar and Schwartz (2003) include a third factor in their model to account
for the term structure of oil prices. The oil price is the common example those
general commodity models are applied to. Nevertheless, as explained above
these models work for coal prices as well.
The trade of emission allowances started only a few years ago and the trading
mechanisms are still under development (compare Section 2.5). Therefore,
there are only a few publications of stochastic models so far. Daskalakis et al.
(2009) extend the common Geometric Brownian motion by a jump-diffusion
component. A stochastic equilibrium model is formulated by Seifert et al.
(2008). Benz and Trück (2009) compare regime-switching approaches and a
GARCH model for the spot price of emission allowances.
Examples of modeling approaches using cointegration are given by de Jong and
Schneider (2009) and Paschke and Prokopczuk (2009). de Jong and Schneider (2009) analyze the dependencies in various European gas markets as well
as the dependency of gas and power prices in the futures market. Paschke
and Prokopczuk (2009) analyze the cointegration of crude oil, heating oil and
gasoil.
5.2. Three-dimensional random walk
The most basic modeling approach for a discrete univariate non-stationary
time series is a random walk
Xt = Xt−1 + εt
with εt ∼ W N (0, σ 2 ). Only the parameter σ needs to be estimated. This
concept may be extended to the multivariate case where dependencies between
the variables are present. For the multivariate process Xt = (X1t , . . . , XN t )0
Xt = Xt−1 + εt
75
with εt ∼ N3 (0, Σ) describes the multivariate random walk. The covariance
matrix Σ contains the variances of the single variables as well as the dependencies between the variables.
The random walk is the discrete equivalent to the Brownian motion. If a
Brownian motion is applied to logarithms of a variable Xt then the stochastic
process Yt = exp (Xt − σ 2 t/2) is a Geometric Brownian motion. Especially, in case of price variables the (multivariate) random walk is usually
applied to logarithms. This transformation ensures simulation results greater
than zero. Nevertheless, the resulting multivariate Geometric Brownian motion contains the correlations between the variables.
In our case the discretized three-dimensional Geometric Brownian motion is
applied to the prices of oil, coal and emission allowances. We write
  
 

1t
ln Ot−1
ln Ot
ln Ct  = ln Ct−1  + 2t 
3t
ln Et−1
ln Et
with (1t , 2t , 3t )T ∼ N3 (0, Σ), the three-dimensional normal distribution with
mean 0 and covariance matrix Σ. The drift parameter µ is excluded from our
considerations. Possible trends of price variables are included by means of the
future contract prices as described in Section 3.5.
After including recent prices in the futures market the scenarios generated by
this model represent the historical volatility as well as the expected level of
the futures market (see Figure 5.1).
5.3. Correlated two factor model
A more sophisticated modeling approach in comparison to the Geometric
Brownian motion was proposed by Schwartz and Smith (2000). We give a
short overview of their model and present the multivariate extension to be
applied to our multi-commodity situation.
The logarithm of the spot price of a commodity is decomposed into two factors
ln St = χt + ξt .
The factor (ξt ) is called equilibrium price level and is modeled by a Brownian
motion with drift. The short term deviations (χt ) are covered by an AR(1)
76
Figure 5.1.: Historical prices of oil (top), coal (center) and emission allowances (bottom) from January 2009 till September 2012. Ten
realizations of the three-dimensional random walk for October
process, the discrete equivalent of an Ornstein-Uhlenbeck process.
χt = αχt−1 + z1t
ξt = µξ + ξt−1 + z2t
The white noise processes (z1t ) and (z2t ) with variances σχ2 and σξ2 have correlation ρ. This model setup leads to a log-normal distribution for the spot
price with parameters given by Schwartz and Smith (2000).
To derive the risk-neutral model two additional parameters are included. λχ
and λξ reduce the drifts of both processes which is a standard way of setting
up risk-neutral models. The model can be stated as
χt = −λχ + αχt−1 + ε1t
ξt = µξ − λξ + ξt−1 + ε2t
with Corr (ε1t , ε2t ) = ρ. (λχ , λξ , µξ , α, ρ, σξ , σχ ) are the unknown parameters of
this model. The parameters are estimated using the Kalman filter. For this
purpose the model is formulated in state space form. The state space form
77
consists of the transition equation
xt = c + Gxt−1 + wt
with xt = (χt , ξt )T , E(wt ) = 0, Var(wt ) = W = Cov(χ∆t , ξ∆t ) and
−κ∆t −λχ ∆t
e
0
c=
,G =
.
(µξ − λξ ) ∆t
0
1
∆t is the time step. κ = 1 − α is the speed of mean reversion in the AR(1)
process (χt ). As the variable xt is not observable the measurement equation
y t = dt + F t x t + v t .
including observable futures prices is required. The log future contract prices
with times to maturity T1 , . . . , Tn in the vector yt = (ln FT1 , . . . , ln FTn )T are
the basis for the estimation. For simplification we use T1 , ..., Tn instead of the
more precise notation T1 (t), . . . , Tn (t).


e−κT1 1

.. 
Ft =  ...
.
e−κTn 1
and dt = (A(T1 ), . . . , A(Tn )) with
λχ
A(T ) = (µξ − λξ ) T − 1 − e−κT
κ
2
σ
1
χ
2
−κT ρσχ σξ
−2κT
+
+ σξ T + 2 1 − e
1−e
2
2κ
κ
are further parameters of the measurement equation. Like (wt ), (vt ) are serially uncorrelated normally distributed innovations with mean zero and covariance matrix Cov(vt ) = V. V needs to be estimated as well.
The above is a spot price model consisting of a short term and a long term
factor. A suitable method for parameter estimation is given and historical
data is available. For example, prices of various monthly future contracts can
be chosen in case of oil. Given this historical data short term and long term
component can be separated and modeled as described above.
Extension to three-dimensional case
In the following we extend the model to a multi-commodity situation where
every single commodity is modeled by the approach from above regarding the
correlation structure.
78
The risk-neutral three-dimensional model is formulated as
χit = −λiχ + αi χit−1 + εi1t
i
+ εi2t
ξti = µiξ − λiξ + ξt−1
where i ∈ {Coal, Oil, Emission allowances} = {C, O, E} = A and the correlations Corr (εi1t , εi2t ) = ρi . The notations for the state space form can be
derived from the notations above. The upper index i is added to denote the
corresponding commodity. This means that we derive the state space form
with transition equation
 C  C  C  C   C
xt
c
G
xt−1
wt
O
O  O 
x O




xt−1 + wtO 
= c
+ G
t
xE
cE
GE
xE
wtE
t
t−1
and measurement equation
 C  C  C  C  C
vt
xt
Ft
dt
yt
O
O  O



ytO  = dO
v
x
F
+
+
t
t
t
t
vtE
xE
FE
dE
ytE
t
t
t
where we have the obvious analogy to the form above. In the measurement
equation we allow for different numbers of futures prices to be available. Even
the times to maturity may differ. As a consequence all vectors and matrices
in the measurement equation have m = nC + nO + nE rows. For example, the
T
i
i
i
analogon to yt is yt = ln FT i , . . . , ln FT i
for i ∈ A.
1
ni
The reason for a three-dimensional model instead of three independent models
is given by the correlations between the variables. These correlations can be
found in the errors of the transition equation. This (6 × 1) vector of errors
from short term and long term processes of each commodity contains the dependencies between the different model components. We refer to this vector
as (wt ) as in the one-dimensional case.
The covariance matrix of (wt ) is one of the unknown model parameters. Estimating Cov(wt ) means estimating 21 parameters (six variances and 15 covariances). On the one hand, this high number of parameters might cause
numerical problems for the Kalman filter. On the other hand, not all of the 15
covariances are needed. Remembering that the six elements of (wt ) represent
three short term and three long term factors for the three commodities we can
assume several covariances to be zero. There is no fundamental reason for any
short term process of a commodity to be correlated with a long term process
of another commodity. Furthermore, we do not allow for any correlations between the short term processes. The common movements of commodity prices
79
are supposed to be covered by the correlated long term processes but short
term price movements are assumed to be independent. For example, political
instability in an oil producing country will not affect the coal price as strong
as the oil price. Hence, short term deviations can be assumed independent.
As a result of the considerations above the covariance matrix can be described
as


W11 W12 0
0
0
0

W22 0 W24 0 W26 



W33 W34 0
0 


W = Cov (wt ) =  ..
.
 .

W
0
W
44
46



W55 W56 
...
W66
This choice reduces the parameters to be estimated from 21 to 12. The assumptions taken above have a fundamental justification but might not be supported
by the data. This means that the estimation of all 21 parameters would not
result in nine parameters being equal to zero. The estimates might be significantly different from zero by accident. Therefore, it is not possible to rely on
the parameter estimation to incorporate these fundamental considerations.
A further simplification is made in the parameter estimation of V = Cov(vt ).
The errors (vt ) cannot be derived from the original model consisting of the
short term and long term factor but are included in the state space form. The
variables can be interpreted as measurement errors or errors of the model’s fit
to observed prices. As in Schwartz (1997) and Schwartz and Smith (2000) we
assume these errors to be uncorrelated. Hence, the covariance matrix V is assumed to be diagonal. This results in a reduction of the number of parameters
from (m + 1)m/2 to m.
In total, the estimation of the parameters λiχ , λiξ , µiξ , αi , W, V with i ∈ A is
required for the three-dimensional model. Due to the simplifications of W and
V the model has 24+m parameters. Each one-dimensional model contains nine
parameters. As the mean reversion parameters, risk premiums and variances
of the three-dimensional model are already included in the one-dimensional
model the independent models can be estimated first. The parameters from
this estimation can be used as starting values for the parameter estimation in
the correlated model. This reduces the problem of finding adequate starting
values for the correlated model to the problem of finding three sets of starting
values for the independent models.
Initial estimates of mean and covariance matrix can be derived from the historical data. For the speed of mean reversion there is no heuristic approximation
80
but at least it is known that the parameter is in the interval (−1, 1). This is
guaranteed due to the decomposition into a non-stationary and a stationary
term. Thus, we are able to generate a set of adequate starting values to ensure
convergence of estimation by means of the Kalman filter.
After including recent prices in the futures market (compare Section 3.5) the
scenarios generated by this model represent the historical volatility as well as
the expected level of the futures market (see Figure 5.2).
Figure 5.2.: Historical prices of oil (top), coal (center) and emission allowances (bottom) from January 2009 till September 2012. Ten
realizations of the correlated two factor model for October 2012
till September 2013.
The estimated correlations between the long term factors within this model
are given in Table 5.2. The comparison to Table 5.1 reveals a lower correlation
between prices of oil and emission allowances. The estimated correlation of
0.02 is closer to the expectation of no correlation between those variables.
As explained above, there is no fundamental reason for a strong correlation
between these time series. The other two correlations are close to the empirical
ones. This gives support to the model as it is capable of reproducing the
correlations of the input variables.
81
Oil
Coal
CO2
Oil
1
0.36
0.02
Coal
0.36
1
0.21
CO2
0.02
0.21
1
Table 5.2.: Estimated correlations of the long term factors in the correlated
two factor model for the prices of oil, coal and CO2 emission
allowances from 2008-2012.
5.4. Cointegration
All time series can be classified with respect to the property stationarity. While
mean and variance of a stationary time series are independent of the time nonstationary time series exhibit deterministic or random fluctuations of mean
or variance. Linear combinations of any stationary time series are stationary
as well. This result does not even require correlation between the variables.
In contrast, the linear combination of non-stationary time series is usually not
stationary. This can easily be seen in an example. The typical process for nonstationary time series is a Geometric Brownian motion. The difference or any
other linear combination of two Geometric Brownian motions is non-stationary
if their log returns have different variances. The result holds even in case of
correlation 1 between the log returns of both variables. Vector processes for
differenced variables do not change this situation.
The issue of stationary linear combinations of non-stationary variables is relevant as many practical examples exhibit this constellation. Such variables
might contain the same non-deterministic trends but do not move to far apart.
For example, prices of crude oil, gasoil and fuel oil are non-stationary. Refineries use crude oil to produce gasoil and fuel oil. As the costs for refineries
are nearly constant the spread between crude oil, gasoil and fuel oil (the crack
spread) is not subject to strong fluctuations. An increase of the crack spread
would lead to new refineries reducing the spread and vice versa. Fluctuations
of the spread, e.g. due to refinery outages, are comparably low. Thus, the crack
spread is stationary. A well-known example concerning electricity prices is the
clean dark spread describing the costs for a coal-fired power plant (compare
Section 6.3). This spread is relevant for the valuation of such power plants.
Thus, an adequate model is needed.
The solution to such problems is given by the concept of cointegration as first
proposed by Granger (1981). The major result making cointegration applicable
to practical situations is the Granger representation theorem given by Engle
and Granger (1987). In the following we present the concept. This includes the
82
5.4. Cointegration
mathematical basics as well as adequate procedures for testing and parameter
estimation. The results are based on Johansen (1995). Finally, the concept is
applied to the prices of coal, oil, gas and emission allowances.
5.4.1. Definitions and properties of cointegration
In the following we consider a set of N variables Xt = (X1t , . . . , XN t )T with autocorrelations for each variable. Additionally, each variable is correlated with
at least one other variable. This is a common situation for financial time series.
Especially, the second characteristic makes the comfortable case of independent models for all variables impossible. The typical model for autocorrelated
univariate time series is an autoregressive model. The multivariate equivalent
is the class of vector autoregressive processes.
Definition 5.1 (VAR(k) process). The N -dimensional process (Xt ) defined
by
Xt = Π1 Xt−1 + · · · + Πk Xt−k + εt ,
for Πi ∈ RN ×N , i = 1, . . . , k,
(5.1)
is called vector autoregressive process of order k. The innovations are independent identically distributed, εt ∼ NN (0, Ω). Initial values X−k+1 , . . . , X0
are given.
The above definition can be extended by a deterministic term consisting of a
constant, a linear term or indicator variables. As the time series throughout
this work have mean zero when VAR(k) processes are applied the deterministic
term is omitted in the definitions. The matrix Ω contains variances of the
components of (Xt ) as well as their covariances.
Further characteristics of the process can be described by means of the characteristic polynomial
A(z) = IN −
k
X
Πi z i ,
for z ∈ R.
i=1
Due to the recursive structure of a VAR(k) process (Xt ) can be expressed in
terms of initial values and innovations.
Theorem 5.2. A VAR(k) process can be written as
Xt =
k
X
s=1
Ct−s (Πs X0 + · · · + Πk X−k+s ) +
t−1
X
Cj εt−j
j=0
83
where C0 = IN and the coefficients Cs ∈ RN ×N are defined recursively by
min(k,s)
Cs =
X
Cs−j Πj ,
s = 1, 2, . . .
j=1
P∞
n
The generating function C(z) =
n=0 Cn z has a radius of convergence δ.
For z ∈ R with |z| < δ the relation C(z) = A(z)−1 holds.
So far, the considered processes are unrestricted allowing for stationary processes as well as non-stationary processes. The behavior of the characteristic
polynomial tells about the stationarity of (Xt ).
Theorem 5.3. Assume that |z| > 1 or z = 1 follows from |A(z)| = 0. Then,
|A(1)| =
6 0 is a necessary and sufficient condition for the stationarity of (Xt ).
Furthermore, the process has the moving average representation
Xt =
∞
X
Cn εt−n
n=0
with C0 (z) =
and z ∈ R.
P∞
n=0
Cn z n = A(z)−1 converging for |z| < 1 + δ for some δ > 0
Parameters of VAR models can be estimated using the maximum likelihood
method. After this estimation the stationarity of the process can be analyzed
according to the conditions given in the theorem above. Usually the time
series are tested for stationarity before the VAR model is estimated. If the
test states non-stationarity the time series is differenced to obtain a stationary
time series. Afterwards, the parameters are estimated.
P
Definition 5.4.
1. A stochastic process (Xt ) satisfying
Xt = ∞
i=0 Ci εt−i
P∞
is called stationary, denoted as I(0), if C = i=0 Ci 6= 0.
2. A stochastic process (Xt ) is called integrated of order d, denoted as I(d),
if (∆d Xt ) is I(0).
This definition of integration implies that multivariate processes consisting of
time series with different orders of integration are integrated. The order is the
maximum of all integration orders of the components.
The major result for cointegration is the Granger representation theorem
given by Engle and Granger (1987). In the following the theorem is presented
based on Johansen (1995). This includes a different formulation of the Granger
representation theorem.
84
5.4. Cointegration
Definition 5.5 (Cointegration). A stochastic process Xt = (X1t , . . . , XN t )T is
cointegrated of order (d, b), if
(i) Xit ∼ I(d) ∀ i = 1, . . . , N , and
(ii) a vector β ∈ RN with β T Xt ∼ I(d − b) exists for b > 0.
Cointegration of order (d, b) is denoted as Xt ∼ CI(d, b). β is called cointegrating vector. The rank of cointegration is the number of linear independent cointegrating vectors.
So far, a class of multivariate time series models (VAR) and a concept to
incorporate dependencies between various variables (cointegration) have been
introduced. The missing part is a method to set up multivariate models with
cointegrated variables. For this purpose the VAR representation in Equation
(5.1) is written in the error correction form
∆Xt = ΠXt−1 +
k−1
X
Γi ∆Xt−i + εt
(5.2)
i=1
P
P
P
where Π = ki=1 Πi −IN and Γi = − kj=i+1 Πj . Let Γ = IN − k−1
i=1 Γi . Using
this parametrization the connection between cointegration and autoregressive
models in VAR or error correction form can be stated. For this purpose we
define the matrix β ⊥ ∈ RN ×(N −r) of full rank such that β T β ⊥ = 0. Technical
details for the construction of β ⊥ are given by Johansen (1995).
Theorem 5.6. If |z| > 1 or z = 1 follows from |A(z)| = 0, and rank(Π) =
r < N , then there exist matrices α, β ∈ RN ×r of rank r such that
Π = αβ T .
A necessary and sufficient condition for the stationarity of (∆Xt ) and (β T Xt )
is
T
α⊥ Γβ ⊥ 6= 0.
Then, the process (Xt ) can be expressed as
Xt = C
t
X
εi + C1 (L)εt + A
i=1
where β T A = 0 and C = β ⊥ (αT⊥ Γβ ⊥ )−1 αT⊥ ∈ RN ×N . Hence, Xt ∼ CI(1, 1)
with cointegrating vectors β. For the function C1 (·) the equation
A−1 (z) = C
1
+ C1 (z),
1−z
z ∈ R \ {1},
85
holds. The radius of convergence for the power series C1 (z) is 1 + δ for some
δ > 0.
This is a different formulation of the well-known Granger representation theorem by Engle and Granger (1987). Johansen (1995) state the above version for
practical reasons. Usually an autoregressive model is applied to data. Using
the above version of the theorem it can be told from the estimated parameters
whether the process is cointegrated or not. The original theorem starts with
cointegration and infers the representation in VAR or error correction form.
The condition αT⊥ Γβ ⊥ 6= 0 is satisfied for all parameters except of a null set,
if the variables are I(1). As the variables in this work have orders of integration
less than or equal to one this condition is usually satisfied.
A major issue for such a concept as cointegration is the estimation of parameters. A maximum likelihood method for parameter estimation and a test for
cointegration are presented in the following.
5.4.2. Parameter estimation and testing
The following explanations focus on testing and parameter estimation for I(1)
variables.
Definition 5.7. The model H(r) of I(1) variables is defined as the submodel
of the VAR in Equation (5.1) where
Π = αβ T
holds for (N × r) matrices α and β. The vector error correction model (5.2)
turns into
k−1
X
∆Xt = αβ T Xt−1 +
Γi ∆Xt−i + εt
i=1
with unrestricted parameters (α, β, Γ1 , . . . , Γk−1 , Ω).
Using this definition models with different ranks of cointegration can be seen
as nested models since
H(0) ⊂ H(1) ⊂ · · · ⊂ H(N )
holds. H(0) implies Π = 0 which corresponds to the usual VAR model in
differences. In contrast H(N ) corresponds to the usual VAR model in VECM
form.
86
5.4. Cointegration
A H(r) model in VECM form can be rewritten as
Z0t = αβ T Z1t + RZ2t + εt
with the notations Z0t = ∆Xt ∈ RN , Z1t = Xt−1 ∈ RN , R = (Γ1 , . . . , Γk−1 ) ∈
T
RN ×N (k−1) and Z2t = ∆XTt−1 , . . . , ∆XTt−k+1 . R is a matrix of parameters
corresponding to the vector Z2t ∈ RN (k−1) . For this model the log-likelihood
function neglecting a constant can be stated as
1
log L (R, α, β, Ω) = − n log |Ω|
2
n
X
T
1
−
Z0t − αβ T Z1t − RZ2t Ω−1 Z0t − αβ T Z1t − RZ2t .
2 t=1
The first order condition for the estimation of R is
n X
b 2t ZT = 0.
Z0t − αβ T Z1t − RZ
2t
t=1
The definition
n
1X
Mij =
Zit ZTjt ,
n t=1
i, j = 0, 1, 2,
with the property Mij = MTji allows for the reformulation of the first order
condition above:
⇔
b 22
M02
= αβ T M12 + RM
b (α, β) = M02 M−1 − αβ T M12 M−1 .
R
22
22
As a consequence the residuals obtained by regressing (Z0t ) and (Z1t ) on (Z2t )
can be defined as
R0t = Z0t − M02 M−1
22 Z2t ,
R1t = Z1t − M12 M−1
22 Z2t .
This leads to the concentrated likelihood function
n
T
1X
1
R0t − αβ T R1t Ω−1 R0t − αβ T R1t .
log L (α, β, Ω) = − n log |Ω|−
2
2 t=1
The regression equation
R0t = αβ T R1t + e
εt
87
would lead to the same likelihood function. This is a reduced rank regression as described by Anderson (1951). With the definition
n
1X
Rit RTjt = Mij − Mi2 M−1
Sij =
22 M2j ,
n t=1
i, j = 0, 1,
estimators for α and Ω can be obtained. For fixed β the regression equation
above is linear and the estimators are
−1
b
α(β)
= S01 β β T S11 β
and
−1
T
b
b
b
Ω(β)
= S00 − S01 β β T S11 β
β T S10 = S00 − α(β)
β T S11 β α(β)
.
The estimators for α, Ω and R deduced above depend on β. Replacing α
b
b
and Ω by α(β)
and Ω(β)
in the concentrated likelihood function results in
a likelihood function only depending on β. With some more rather technical
considerations the maximum likelihood estimator for β can be derived.
Theorem 5.8. Given the hypothesis
H(r) : Π = αβ T ,
the maximum likelihood estimator of β is the result of the following procedure.
The equation
λS11 − S10 S−1
00 S01 = 0
b1 > · · · > λ
bN > 0 with the corresponding
is solved for the eigenvalues 1 > λ
b = (b
b T S11 V
b =
bN ). The eigenvectors are normalized by V
eigenvectors V
v1 , . . . , v
IN . The cointegrating relations β result from the eigenvectors
b = (b
br ) .
β
v1 , . . . , v
This estimator of β is inserted in the equations above to derive estimates of
α, Ω and R. The maximized likelihood function for the model is given by
L−2/T
max (H(r))
= |S00 |
r Y
bi .
1−λ
i=1
The likelihood ratio test statistic Q(H(r)|H(N )) for H(r) in H(N ) is given by
−2 log Q (H(r)|H(N )) = −n
N
X
bi .
log 1 − λ
i=r+1
The statistic changes to
br+1
−2 log Q (H(r)|H(r + 1)) = −n log 1 − λ
88
5.4. Cointegration
for testing H(r) in H(r + 1). Under the assumption rank (Π) = r the asymptotic distribution is
)
(Z
Z 1
−1 Z 1
1
w
BBT du
B(dB)T
−2 log Q (H(r)|H(N )) −→ tr
(dB)BT
0
0
0
where B is a N − r dimensional Brownian motion.
A more detailed deduction of this theorem, the proof and simulated values of
the distribution can be found in Johansen (1995).
The above theorem and the previous deduction introduce a procedure to estimate all parameters for the cointegrated model with a given rank of cointegration r ∈ {0, . . . , N }. A last open question to be answered is the determination
of r.
The rank of cointegration r takes on the values 0, 1, . . . , N . Due to the nested
structure of the models with different ranks of cointegration a sequence of tests
can be run. The test statistic Qr = −2 log Q (H(r)|H(N )) is compared with
its quantile cr . If Qr < cr , then rb = r. If Qr ≥ cr then the test statistic and the
quantile are compared for r + 1. The quantiles cr result from simulated values
of the distribution of the test statistic as mentioned above. After the procedure
is carried out for r = 0, . . . , N the highest r satisfying the above condition is
known. Thus, the parameters of the model H(b
r) can be estimated.
After introduction of cointegration including a test for cointegration, parameter estimation and determination of the rank of cointegration the concept may
be applied. In the following we present the application to the prices of oil,
coal, gas and emission allowances.
5.4.3. Application to commodity prices
The application of cointegration procedures to real data implies various difficulties.
• The available period of data might be too short to identify a cointegrating
relationship. Cointegration describes a common long term movement
that might not be obvious in a short data set.
• The available period of data might contain too many "disturbances" by
exceptional events to identify a cointegrating relationship. A war in oil
89
producing countries leads the oil price away from its equilibrium price
described by the cointegration.
• The chosen products might be too volatile to identify a cointegrating
relationship. Contracts on the future markets with long maturity are
less volatile than short term products.
As our cointegrated commodity price model is supposed to be an input factor
for the spot price of electricity we are interested in cointegration on "spot"
products. This means front month contracts in our case. As mentioned above,
the front month contracts might be too volatile. If we use long term future
contracts instead there is a better chance to identify a strong cointegration.
To use this, another process linking long term future contracts to front month
contracts is needed. In order to avoid this additional complexity of the model
we stay with prices of front month contracts.
Based on the data used in this work the front month prices of oil, coal and
emission allowances are not cointegrated. The test by Johansen (see Section
5.4.2) cannot state cointegration at a reasonable level of confidence. This
result is not surprising if we consider Figure 5.3. The prices of oil and coal
have a common trend but the prices of emission allowances are independent.
Inclusion of natural gas as a fourth variable changes the situation. The prices
of gas and emission allowances show a similar (though mirrored) behavior.
The cointegration test states this set of variables to be cointegrated of rank
one at a significance level of 0.05. Cointegration of rank three can be stated
for the same data set at a significance level of 0.1. If log prices are used instead
the cointegration of rank one is identified at the same significance level. The
cointegration of rank three cannot be stated anymore. The drawback of a
model based on original prices is the chance of negative prices in simulations.
Especially, prices of emission allowances are close to zero which immediately
leads to negative simulation results. Therefore, we decide for the cointegration
of rank one based on log prices.
Oil
Oil
1
Coal 0.36
Gas 0.00
CO2 0.10
Coal
0.36
1
0.57
0.38
Gas CO2
0.00 0.10
0.57 0.38
1
0.03
0.03
1
Table 5.3.: Correlations of 1000 realizations of prices of oil, coal, natural gas
and emission allowances based on the cointegration approach.
90
5.4. Cointegration
Figure 5.3.: Historical front month contract prices of oil (blue, Euro per barrel), coal (green, Euro per ton), emission allowances (red, Euro
per ton) and gas (light blue, Euro per MWh) from January 2009
till September 2012. The mean is subtracted.
The simulated correlations of the cointegrated model given in Table 5.3 deviate from the empirical ones given in Table 5.1. The greatest deviation is
observable in the correlation of coal and gas prices: 0.57 (cointegration) to
0.35 (empirical). Not even this deviation is big enough to challenge the approach. The concept of cointegration describes a stronger relationship than
the (linear) correlation of variables. As this relationship is statistically significant it is more important to include the cointegration than to match the
empirical correlations. Two realizations of the cointegration model are given
in Figure 5.4. The strong connection of coal and oil prices within the model
makes both variables move together. Gas and emission allowances show the
typical fluctuations within a certain price range.
91
Figure 5.4.: Two realizations of the cointegration approach for oil (blue), coal
(green), emission allowances (red) and natural gas (light blue).
The time period from October 2012 till September 2013 is simulated.
92
6. Discussion of electricity price
models
In the previous chapters a new spot price model for electricity based on various
fundamental input variables is introduced. Regarding the load and the commodity prices several alternative modeling approaches are presented. In this
chapter we give a discussion of all approaches presented above with respect
to the resulting electricity prices. This includes the comparison to existing
models from the literature. Before starting the comparison a short overview
of the presented models is given.
6.1. Overview of presented models
The spot price process is defined as
(S)
St = g (lt , Gt , Ct , Et , t) + Xt
which was defined more general in Equation (3.2). The function g can be
obtained by applying the model to the whole data at once and by applying it
to each of the 24 hourly time series. We distinguish these approaches in our
analysis.
Model E1: The model
g (lt , Gt , Ct , Et , t) = a0 + a1 1Sat (t) + a2 1Sun (t) + a3 lt + a4 Gt 1A (t)
+a5 Ct + a6 Et + a7 lt Gt + a8 lt Ct + a9 lt Et
is applied to the whole data set (compare Equation (3.5)). 1A (t) describes
the hours from 7 a.m. to midnight.
Model E2: The model
g (lt , Gt , Ct , Et , t) = a0 (t) + a1 (t)1W e (t)
+a2 (t)lt + a3 (t)Gt + a4 (t)Ct + a5 (t)Et
+a6 (t)lt Gt + a7 (t)lt Ct + a8 (t)lt Et
93
is applied to the 24 hourly time series (compare Equation (3.6)). The
indicator variables for Saturday and Sunday are aggregated to one indicator variable for the whole weekend.
There are two approaches to obtain the residual load process (lt ).
Model L1: The grid load (see Equation (3.8)) is modeled as
bt + Xt(L) + Yt(L) .
Lt = L
The renewables, being subtracted from the grid load, are obtained by
means of a historical simulation with varying forecasts of the future capacities as described in Section 3.4.1.
Model L2: In analogy to the grid load, the residual load (see Equation (3.9))
is directly modeled as
(l)
lt = b
lt + Xt .
For the commodity prices we consider the following three model alternatives.
Model F1: The prices of oil, coal and emission allowances follow a threedimensional random walk (see Section 5.2)
  
 

1t
ln Ot−1
ln Ot
ln Ct  = ln Ct−1  + 2t  .
3t
ln Et−1
ln Et
The gas price model with normalized cumulated heating degree days (Λt )
and oil price component (Ψt ) as proposed in Equation (4.2)
bt + α1 Λt + α2 Ψt + Xt(G)
Gt = G
is used.
Model F2: The prices of oil, coal and emission allowances are modeled by
the correlated two factor model described in Section 5.3. The gas price
model is the same as in F1.
Model F3: The log prices of oil, coal, emission allowances and gas are cointegrated of rank one. The model is simulated using the VAR representation
in Equation (5.1).
For details about the stochastic processes we refer to the corresponding chapters where the calibration of these processes is presented. This overview focusses on the relationships between the model components. In the following
94
6.2. Analysis of model features
the spot price models are compared with respect to the underlying load and
commodity price models.
A comprehensive discussion of the above models with respect to statistical
criteria is not possible for several reasons.
• Typical criteria for model selection such as the AIC criterion are not
applicable as likelihood functions for the models are not available.
• The regression models E1, E2, L1 and L2 can be compared according
to their R2 or adjusted R2 . Nevertheless, the R2 cannot tell whether the
combination of E1 and F1 provides a "better" model than the combination E1 and F2 as the commodity price models are no regression models
where the coefficient of determination can be calculated.
• The data contains structural changes due to the increase of renewables
and the decrease of nuclear capacities. Therefore, it is not advisable to
divide the data used in this work into two sets: one for model calibration
and one for model validation. The data used for model validation would
consist of the most recent data, i.e. the data containing the effects of
changed nuclear and renewable capacities. Thus, the model cannot learn
about these effects from the remaining data.
For these reasons a final decision about the best model cannot be based on
a statistical analysis. In the following, we discuss the models with respect to
various aspects.
Comparison of R2
The models E1 and E2 can be compared according to the goodness-of-fit
R2 . This measure can be calculated for some other models as well. Table
6.1 gives an overview of some models. The value belonging to SMaPS refers
to the goodness-of-fit of the model introduced by Burger et al. (2004). This
model is one of the first fundamental spot price models (compare Section 3.2
and 3.3). Not even the use of residual load instead of grid load remarkably
increases the goodness-of-fit of the SMaPS model. The fuel-adjusted heat-
95
R
2
E1
0.69
E2 SMaPS
0.71
0.35
Andersson et al.
0.79
Table 6.1.: In-sample comparison of different models with respect to R2 .
rate model proposed by Andersson et al. (2013) exceeds the R2 of the other
models. R2 = 0.79 is stated in their paper based on data from January 2010
till December 2010. Fitting a model to this data leads to an increased R2
due to the smaller number of observations. Furthermore, influences of the
economical crisis 2008/2009, the decrease of nuclear capacities and the increase
of renewables do not affect this data.
Stylized facts
The proposed models are supposed to exhibit the stylized facts given in Section
3.1.
• All models compared above have the load as an input variable. Therefore,
the seasonalities of the load are reflected by the resulting spot price
processes.
• The SMaPS model cannot reproduce negative prices due to the multiplicative structure of the model. The fuel-adjusted heat-rate model is
based on a dual-exponential model which allows for negative prices as
well. Our models have an additive structure and can generate scenarios
including negative prices.
• The SARIMA process of the SMaPS model relies on normal distributed
innovations. The residuals of the fuel-adjusted heat-rate model are white
noise of normal distribution. Although the dual-exponential function
strongly increases it is not guaranteed that price spikes occur. Thus,
these models do not simulate price spikes. Our models use a SARIMA
process with t-distributed innovations. Due to the low number of degrees
of freedom this distribution has heavy tails and is capable of generating
price spikes.
• The SARIMA processes of our models and the SMaPS model are meanreverting like the white noise process in the fuel-adjusted heat-rate
model.
• The white noise of the model by Andersson et al. (2013) does not consider
96
the autocorrelations of the residuals. This feature is covered by the
SARIMA process included in our model.
Input variables
The SMaPS model relies on the grid load adjusted by average availability
of power plants. Apart from pricing electricity derivatives the model can be
applied for pricing of retail power contracts (see Chapter 7). Our models E1
and E2 are capable of risk-adequate pricing on the retail market, if the load
model L1 is used. Adequate valuation of conventional power plants is not
possible due to the missing commodity prices in the SMaPS model. The fuel
adjustment of Andersson et al. (2013) considers the prices of coal and emission
allowances. Natural gas or oil prices are neglected. Thus, the model can be
used for the valuation of coal-fired power plants but not for valuation of gasfired power plants or pricing on the retail market. In addition to the residual
load, the prices of coal, gas and emission allowances are included in our model.
Altogether, this allows for comprehensive valuations of power plants and riskadequate pricing on the retail market. The model can easily be extended by an
oil price component for valuations of oil price related assets or derivatives.
Long term uncertainty
Burger et al. (2004) included a long term process (random walk with drift)
in their model to account for the uncertainty of price development on a long
term horizon. Such a factor is not mentioned by Andersson et al. (2013) but
for risk management purposes this uncertainty plays a major role. Reasons for
changes of electricity price level on a long term horizon are changing commodity
prices, changes of generation capacities and changes of demand. Most of these
reasons are already considered in our models. The non-stationary commodity
price models reflect the uncertainty of commodity price levels. In contrast
to model L2, the grid load model in model L1 includes a long term factor
to account for possible changes of the load level for economic reasons. The
models for renewables have capacity forecasts as a major component. Except
of capacity changes of conventional power plants this source of long term price
uncertainty is modeled as well. Altogether, the major part of sources of long
term uncertainty is covered by the model components. Another stochastic long
term factor is unnecessary.
97
Load model
Model L1 is favored over model L2 as the use for retail pricing as presented
in Chapter 7 is not possible with model L2. Furthermore, expectations about
future capacities of renewables can only be included in model L1. Generation
by renewables is the major driver of the residual load in the years to come.
Choosing model L1 gives the chance to achieve some further improvements
with more sophisticated models for generation by renewables. The historical
simulation used so far is a simple approach that needs to be replaced by more
reliable models.
6.3. Out-of-sample model validation
In the discussion above we validated the models with respect to required features and input variables. Furthermore, the comparison with two fundamental
spot price models from the literature gives evidence for the quality of the
model. A preference for either of the models E1 and E2 is not justified so far.
The differences between these models are small. The increased R2 of model E2
is achieved by an increased number of parameters. Both models give reliable
results so that both alternatives might be used.
A decision with stronger impact is the choice of the commodity price model.
The models for the non-stationary commodity prices oil, coal and emission
allowances are pure stochastic. Therefore, we cannot measure the in-sample
goodness-of-fit. Comparison of mean (obtained from the prices of recent future prices) and variance (increasing with time) with historical prices is not
meaningful.
The most relevant issue concerning commodity prices is the issue of dependencies. The dependencies on electricity as well as the dependencies among
themselves need to be covered by the model. The relationship between electricity and commodity prices is determined by the choice of the polynomial
approximation. Thus, the dependencies among the commodities themselves
determined by the commodity price model need to be analyzed. We examine
this issue by comparing the simulated clean dark spread to the historical
clean dark spread. The spread is defined as
CDSt = St −
98
a · Ct b · E t
−
.
η
η
a is needed for the conversion from tons to MWh depending on the quality
of coal in use. We decide for an average value of a = 1/7. The same argumentation leads to the factor b describing the emissions intensity per MWh
of generated electricity. For this calculation we set b = 0.33. The efficiency
η describes the percentage of energy content of coal being transformed into
electricity. This percentage strongly depends on the power plant being used.
We assume an average of η = 0.35. The exact choice of these three factors
depends on the power plant being valuated. For comparison of historical and
simulated spreads any choice is possible. Nevertheless, these average values
give realistic spreads.
For an out-of-sample comparison we simulate the clean dark spread on monthly
prices, i.e. the average of simulated hourly or daily prices is taken as the
monthly price. In this example the clean dark spread for January 2013 is
calculated. The historical data used for model calibration ends in September
2012. Based on this data the simulated month is the fourth front month.
Therefore, the clean dark spread of historical prices of the fourth front month
is used for the comparison. As these prices are not used within the model
calibration it is an out-of-sample validation of the model.
The clean dark spreads in Figure 6.1 are centered at the mean of the historical
clean dark spread. These results are obtained by using the commodity price
models F1-F3 in combination with the spot price model E1. Model E2 gives
similar results.
Model
5% 95% Range
Random Walk -4.5 17.7 22.2
Two factor
-5.7 18.3 24.0
Cointegration -2.7 16.1 18.8
Historical
-1.3 15.4 16.7
Table 6.2.: Quantiles of simulated and historical clean dark spreads.
According to the visualization in Figure 6.1 the model F3 is the best approximation of the historical spread. This is supported by the quantiles given in
Table 6.2. The range between 0.05 and 0.95 quantile reveals the close fit of the
cointegrated model to the historical range. The differences to the other models
are remarkable. Though a close fit to the historical spreads is requested the
simulations are supposed to exceed the historical range of the spread. Both
is guaranteed by the cointegrated commodity price model. The other models
exceed the historical range too far.
99
Figure 6.1.: 1000 simulations of the clean dark spreads for January 2013 resulting from model F1 (red), F2 (green) and F3 (black) compared to the historical clean dark spread of the fourth front
month (blue).
Electricity price scenarios
According to the above argumentation it is advisable to use any of the merit
order curve approximations E1 or E2, the grid load model with distinct models
for the renewable generation L1 and the cointegrated commodity price model
F3. This combination of model components leads to the widest range of possible applications. One model for "all" valuations leads to consistent results
which is of particular interest for risk management issues. For this purpose
the inclusion of the oil price in model E1 or E2 is advisable.
In Figure 6.2 realizations of our preferred model combination are presented.
Especially, the scenarios for one week in June reveal a wide range of prices due
to the strong influence of renewables. The lower price level on weekends can
be observed as well.
The scenarios do not reveal a structure as obvious as suggested by the consideration of an HPFC (compare Section 7.2.1). The expectations given by an
HPFC are still valid but due to stochastic influences most realizations of the
100
Figure 6.2.: Simulated spot prices of electricity for 2013 (top). Four scenarios
for one week in June 2013 (bottom). All scenarios are shifted to a
price forward curve based on future contracts from 2012/09/28.
model will not reveal these structures. Grid load, generation of renewables,
commodity prices and the short term process of the spot price cause stochastic
price behavior.
101
7. Pricing of retail power
contracts
In the following we give a detailed description of risk factors in retail power
contracts and their pricing. It is published in Burger and Müller (2011). Section 7.3 is adapted for the spot price model introduced in Chapter 3. The
example gives evidence for the new spot price model being applicable to retail
pricing issues.
7.1. Introduction
Electricity retail markets have changed dramatically since the establishment
of wholesale markets. Before the markets were liberalized, retail prices were
cost-based and were reviewed by a regulator. Since the introduction of the first
wholesale electricity markets, retail prices have been deduced from published
wholesale market prices.
If electricity is sold to an end customer and the sold volume depends on the
consumption of the customer, we refer to this market as a retail market. This
includes business-to-business and business-to-consumer sales. In this work, for
retail contracts, the volume risk and all associated costs (such as balancing
power) are borne by the supplier. In practice, some end customers have contracts that are derived from wholesale contracts. A modular computation of
risk premiums allows the application of the following pricing concepts for these
customers. There are two common types of contracts on the retail market:
fixed-price contracts and indexed contracts. The supply price for fixed-price
contracts is known when the supply contract is concluded. Indexed contracts
contain a specified number of fixing possibilities where the customer has the
chance to diversify the price by fixing the price over several dates. Fixed-price
contracts can be regarded as a special case of an indexed contract with only
one fixing possibility.
103
We consider full-service contracts on the German market with a contract horizon of one year unless otherwise specified. In contrast with the retail market,
the wholesale market includes standardized over-the-counter contracts or takes
place at an energy exchange such as the EEX or Nord Pool in Scandinavia.
Retail electricity contracts differ from standardized wholesale products in several respects. The most significant difference is the uncertain volume of the
contracts. While the volume of a forward or future contract is fixed, the volume of a retail contract depends on the consumption of the customer. This
uncertainty is not an option in the usual sense of financial products because it
is not exercised in a market-oriented fashion. While the subject of the pricing
of electricity on wholesale markets has been widely researched (see, for example, Schwartz (1997), Eydeland and Wolyniec (2003), Geman (2005), Weron
(2006), Burger et al. (2007)), the pricing of electricity retail products is less
commonly considered. Despite this, the pricing of retail electricity is essential
for suppliers, and calculating the risks is not a straightforward task. Difficulties encountered with the pricing of electricity retail contracts include the
lack of publicly available data and the necessity of a consolidated knowledge
of operational market details.
In the literature there are only a few works that name the risk factors involved
in retail contracts: Although, for example, Boroumand and Zachmann (2012),
Lemming (2003), Ojanen (2002) and Unger (2002) describe the sources of risk,
they do not address the issue of pricing the main risk factors, instead using a
risk management-based approach. Comprehensive approaches for retail contract pricing including risk premiums have been proposed by several authors.
For example, Keppo and Räsänen (1999) use stochastic processes for consumer
consumption and energy price. Combining these processes, a future value of
a customer’s consumption pattern is derived and used to price specific call
options. These call options lead to a value for the electricity tariff of an end
customer. In contrast to this option-based approach, Dhaliwal et al. (2001)
introduce a customer preference index to indicate a range of acceptable retail prices. The range of market prices is covered by a fuzzy set approach.
These two components together lead to a retail price customers will be likely
to accept.
A so-called breakeven price is calculated by Eakin and Faruqui (2000): The
breakeven price is specified as the price that does not lead to an expected
profit or loss for the supplier. As log-normal distributions are assumed for
price and customer load and a correlation between these processes is used,
the breakeven price contains the risks of a retail contract. Karandikar et al.
(2007) and Prokopczuk et al. (2007) present some retail pricing methods using
the risk-adjusted return on capital (RAROC) and capital asset pricing model
104
7.1. Introduction
(CAPM) approaches.
A major issue for retailers is the minimization of procurement costs, i.e., the
reduction of settlement risk. Buying electricity with fixed quantities on the
wholesale market and selling flexible quantities to customers exposes retailers to the risk of increased costs due to unfavorable procurement or changing
market prices. Optimization approaches for the determination of optimal procurement quantities, the type of contracts to be used in procurement and
strategies for adjustments within the run of time are presented by various authors. Karandikar et al. (2010), Deng et al. (2005), Bunn et al. (2010), Bunn
et al. (2011), Horowitz et al. (2004) and Deng and Oren (2006) all differ in the
hedging instruments that they consider.
Similar ideas are presented by Arroyo et al. (2007), Balakrishnan et al. (2006)
and Hatami et al. (2009). In these papers the reduction of procurement costs
and settlement risk is treated in one optimization together with the determination of a retail price on the customer side.
One common aspect of major energy utilities having generation assets and retail customers is not reflected in these approaches. Usually the spread between
fuel costs and electricity price is logged to reduce the risk on the generation
side to a minimum. On the retail market these utilities would act like any
other retailer: energy is purchased on the futures market and sold to customers. Therefore, the price for retail customers must not be determined in a
combined optimization of procurement and customer side.
We give a complete overview about the different types of risks in retail contracts. For most types of risk a possible method of determining an adequate
premium is presented. We focus on the uncertainty of the contract volume and
use combined stochastic models for load and price. These models are based
on the models described in Chapter 3.
After this introduction, we explain the decomposition of the total price into
several price components in Section 7.2. This includes the analysis of the various sources of risk and choice of adequate risk measures. The extension of our
spot price models to combined models for price and load is explained in Section
7.3. An application to customer data is used for comparison. The explanations
focus on full-service contracts but the methodology for other contract types
can be derived.
105
7.2. Price components
We decompose the retail prices into three components:
P (t) = F (t) + R(t) + M
which reflect the hourly energy price F (t) deduced from the price on the wholesale market, a possibly time-dependent risk premium R(t) and a sales margin
M . The latter component is the result of a management decision of the supplier
and is not the subject of further analysis in this work. Under a mathematical
point of view, the other components contain a greater potential for further
research. In what follows, the deduction of an hourly price forward curve
(HPFC) from the wholesale market prices (see Section 7.2.1), the sources of
risk within retail contracts (see Section 7.2.2) and some risk measures for the
calculation of risk premiums (see Section 7.2.3) are presented.
7.2.1. Hourly Price Forward Curve
A supplier entering the retail market needs to calculate a fair price of energy in
order to compete with other market participants. The fair price of energy, as
the basic price component, covers expected costs of future delivery of energy
on a specified time granularity. In this work this (finest) granularity is one
hour caused by the fact that this is the finest granularity of spot products in
many countries. The statements can be adapted to any other market situation.
Forward and future contracts often have a large delivery period of a year, a
quarter or a month. So, a method for calculating a forward curve on an hourly
basis needs to be found. Some approaches for the calculation of an HPFC are
proposed by Fleten and Lemming (2003), Koekebakker and Ollmar (2005) and
Burger et al. (2007).
An example of an HPFC for one year is given in Figure 7.1. It is the result of a
simple regression approach based on historical spot prices. This implies the assumption that forward contract prices are expected spot prices. This example
reveals a typical characteristic of forward contract prices: These prices give a
step function whose steps are transferred to the HPFC. Although some of these
steps will realize, it is a more realistic assumption to expect smoother transitions between intervals of the forward contract prices. Therefore, a smoothing
method, as proposed by Benth et al. (2007), might be included in the calculation of the HPFC.
After calculating one price for each hour of a year, the basic price for the
106
Figure 7.1.: Example HPFC for 2010 deduced from monthly and quarterly
forward contract prices from the EEX. A regression model is
used for the calculation of hourly weights from historical spot
prices. Holidays are neglected within this example.
customer is deduced from the HPFC. Usually, in practice, a customer is charged
either one fixed price or one peak and one off-peak price. These peak and
off-peak times might differ from definitions at an exchange. The decision is
left to the supplier. These prices result from averaging the HPFC over the
corresponding hours:
Pt2
basic price =
t=t1 Lt F (t)
Pt2 b
t=t1 Lt
b
(7.1)
bt for t ∈ [t1 , t2 ], which represents the interval of delivwith the load forecast L
ery.
7.2.2. Risk factors
In addition to the costs of delivering energy to the customer, a retail power
contract contains various sources of further costs to the supplier. Some of these
107
factors cause losses on average, while others increase the probability of losses,
but do not contribute to the losses on average. In this section we describe the
most important risk factors with respect to full-service retail contracts.
Price validity period
The process of concluding a retail power contract starts with a request for
a quotation by the customer. The supplier calculates the price quotation, or,
more precisely the basic price, on the basis of the latest forward contract prices.
As soon as this price quotation is submitted, the customer can choose to either
accept or refuse the offer within a certain period of time (the so-called price
validity period). The length of the price validity period ranges from a few
hours to several days, or, in some cases, even some weeks, depending on the
supplier and the customer. If the forward contract prices are increasing, the
customer concludes the contract based on lower forward prices. If, however,
forward prices decrease, the customer demands a new offer based on the actual
lower prices. In a financial sense, the customer is the holder of a call option
on the market price. Since it is unusual to charge an upfront premium for a
price quotation, this option is free of charge for the customer. Including an
option premium in the retail price only means increasing the strike price of
the call option, but this is the only possibility for the supplier, who has to
bear the resulting costs. In theory, as the writer of the option, the supplier
has to receive a premium for giving this optionality. This premium can be
determined using an option-pricing approach (Black-Scholes) or the method
described by Bartelj et al. (2010), although not all customers take advantage of
this optionality. In some cases the decision regarding execution of the option is
determined by duration of decision processes rather than by market prices.
Credit Risk
As soon as a retail contract is signed, the supplier begins sourcing energy
based on a load forecast for the customer. Purchasing the forward contracts
traded on the wholesale market for hedging purposes reduces the price risk
for the supplier. This procedure occupies the risk of the counterpart, i.e., the
customer, defaulting before payment of delivered energy. In case of a defaulting
customer, the utility will realize a loss of the delivered but unpaid energy. The
energy that has not been already delivered can be sold by the supplier again,
but it may not be able to obtain its purchase price. To insure against this
credit risk, the supplier will charge the customer a premium for bearing this
108
risk. The most important components for evaluation of credit risk are the
exposure at default (EAD) and the probability of default (PD). The
EAD is the mark-to-market of the energy amount sold to the customer. The
determination of the PD of a customer is a more complex task for the supplier.
In case of large companies, rating agencies analyze their business activities
and classify the companies. This allows a PD to be calculated. As suppliers
have a portfolio of retail customers, including smaller companies without a
rating from any of the agencies, an internal rating for the estimation of PD is
required. When setting up an internal rating system, calibration of the internal
system on published ratings is essential. This means that the internal system
is calibrated so that external ratings are reproduced as far as possible, if they
are available. The expected losses due to credit risk within a retail contract
can be quantified using EAD and PD.
Price-volume correlation
If the spot price is higher than the normal price level, this is usually the consequence of the grid load being higher than normal. As the grid load is the
sum of all customer loads, there is a high probability of individual customer
loads being above the normal level. Customers are therefore likely to deviate
from the supplier’s forecast in the same direction as the spot price. There is
a price-volume correlation with the following price effect. As soon as a
customer asks for the price quotation it submits data on its past consumption.
This data is used to calculate a load forecast for the delivery period. If the
quotation is accepted, the supplier will begin procurement based on this forecast. Usually the realized load will deviate from the forecast. If the demand
exceeds the previous forecast, the hedged quantity is not sufficient to cover the
demand, and further quantities need to be purchased on the spot market. Due
to the described price-volume correlation, high spot prices are more probable
in this case. This leads to increased sourcing costs for the supplier. Conversely,
surplus quantities resulting from a new, lower forecast have to be placed on the
market. These quantities are likely to be sold at low prices due to the pricevolume correlation. An approach using stochastic dynamic programming to
determine an optimal load forecast is given by Balakrishnan et al. (2004).
Individual volume risk
In addition to these correlated deviations from forecasts, changes in customer
electricity demand might occur for a variety of reasons, such as the extension
109
of the customer’s machine park, or an increase or decrease in the customer’s
production. These incidents can be reduced to individual customers and do not
influence the behavior of all customers. Therefore, this so-called individual
volume risk is not systematically correlated with the price.
Hourly price profile
As described in Section 7.2.1, the basic price is based on the HPFC, which is
an estimation of the future hourly price profile. Within the time to delivery,
the hourly price profile might change. Since market liquidity for hourly profiles
is low, hedging this risk is often not possible. Therefore, the hourly price
profile risk is another component of the total risk. On average, the HPFC
should match the realized price profile, so that the risk does not lead to any
costs. As soon as deviations from the HPFC are realized, additional costs
occur due to the unhedged quantities. Models for the calculation of a risk
premium on hourly price profile risk, individual volume risk and price-volume
correlation is introduced in Section 7.3.
Costs of balancing power
While the demand of any customer differs from the final load forecast, the
transmission system operator has to balance these fluctuations and charge the
supplier for costs of balancing power. Burger et al. (2007) present an approach
for the determination of balancing power costs from historical load and load
forecast data. A model based on combined seasonal autoregressive integrated
moving average (SARIMA) and Markov processes for balancing power market
prices is given by Olsson and Söder (2008). An example of German imbalance
energy prices on a quarter-hourly basis is given in Figure 7.2.
Operational risk
Across the whole process – from giving a price quotation to the delivery of
energy – there are many possibilities for further failure that are aggregated as
operational risk. The duration of the sourcing process and the quality of
load forecast are only two possible sources for negative influences on the total
result. Due to the variety of sources, there is no comprehensive approach for
quantification of operational risk in retail contracts.
110
Figure 7.2.: German imbalance energy prices in July 2010.
The risk factors in this overview can be classified into two groups: systematic
and unsystematic risk factors. While unsystematic risk factors have a zero
mean, systematic risk factors cause additional costs on average. Both groups
contribute to the total risk as there are fluctuations around the mean value.
Therefore, customers are not only charged for the expected costs but also for an
additional risk premium. This premium covers the possibility of losses due to
the underlying risk factors. The classification of risk factors, as shown in Table
Systematic risk
Price validity period
Credit risk
Price-volume correlation
Balancing power
Operational risk
Unsystematic risk
Hourly price profile risk
Individual volume risk
Table 7.1.: Classification of risk factors in retail contracts as systematic or
unsystematic risk factors.
7.1 is fairly theoretical, since a strict differentiation within the calculation of
a risk premium is not always applicable.
While the above-mentioned risk factors are typical for common full-service
111
contracts, there are further types of contracts that reduce some risks. Indexed
contracts, for example, contain a specified number of fixing possibilities, i.e.,
the customer has the chance to diversify the price by fixing the price over
several dates. There is a risk reduction for the customer, who does not fix
the whole quantity at the wrong moment. Nevertheless, common full-service
contracts with fixed prices are still widely used and need to be priced with
respect to the risk. An overview of risk measures for quantification of risk in
electricity retail markets is given in the following section.
7.2.3. Risk-adequate return on capital
While charging the customer for the basic energy price and the expected costs
resulting from the above-mentioned risk factors usually secures a supplier’s
earnings on average, this is not the case in every scenario, nor even in the
majority of them. Therefore, the price of a retail contract has to contain a risk
component as a fair premium for taking on the risk, just as participants in a
financial market ask for a premium to take a risk. In order to cover the risk
of further costs, an adequate risk measure is needed. The concept of charging
customers for a risk premium is comparable with insurance companies charging insurance premiums. A wide range of concepts for calculating insurance
premiums can be found in Goovaerts et al. (1984). We now present the concept
of RAROC as a method for achieving a risk premium. An application of the
RAROC approach to electricity markets is given by Prokopczuk et al. (2007).
A more general discussion can be found in Hallerbach (2004) and Stoughton
and Zechner (2007).
The central idea is that (equity) capital has to be allocated for risks. Most
companies have clear concepts for the return on their (equity) capital. As
(equity) capital is a scarce resource, they only enter contracts if a certain
return on the allocated capital, the so-called hurdle rate, µ, is achieved. The
hurdle rate is the financial objective of each project within a company and
may be influenced by different factors, such as the costs of allocated capital
in case of funding by loans, or the expected return of the shareholders. These
are only two possible approaches for the determination of a hurdle rate among
others, some of which might theoretically be more advanced. As the focus is
on the calculation of risk premiums, the hurdle rate is assumed to be given.
Prokopczuk et al. (2007) give examples of approaches for determining the
hurdle rate. The RAROC approach describes the relation between profitability
and riskiness of a contract:
RAROC =
112
expected return !
≥ µ.
economic capital
7.3. Correlated price-load models
The economic capital, as the capital that needs to be allocated to cover risk
even in worst-case scenarios, is determined by a risk measure based on the
distribution of costs of a retail contract. The aggregated economic capital of all
contracts is supposed to ensure a company’s survival in worst-case scenarios.
Given a certain hurdle rate, the expected return can be determined by the
RAROC approach via
expected return ≥ economic capital · µ.
A common risk measure for determining the economic capital is the valueat-risk (VaR) at a given level of confidence α. Jorion (2006) gives a detailed
discussion of VaR. Using an adequate risk measure in combination with the
RAROC approach leads to an adequate risk premium. After that, the price
can be composed of the basic energy price, the risk premium and, finally, the
sales margin. This theoretical price might differ from the price charged on the
retail market due to competition between suppliers.
As mentioned in Section 7.2.2, the pricing of price-load correlation, individual
volume risk and hourly price profile risk can be carried out simultaneously
using one model. Stochastic price and load scenarios with the correct correlation need to be generated by the model in order to represent the fluctuations
in both components. In the literature, some spot price models using grid
load as a fundamental driver have been proposed (see, for example, Coulon
and Howison (2009), Jermakyan and Pirrong (2008) and Aïd et al. (2009)).
These models generate stochastic price and grid load scenarios but methods
for the deduction of customer load scenarios are not covered. We propose such
a method to extend the spot price model presented in Chapter 3 for retail
pricing purposes.
7.3.1. A customer load model
The spot price as modeled in this work contains the residual load as a fundamental driver. As the grid load is the basis for the residual load the dependency
of spot price and grid load is covered by the proposed model. In order to price
customer load fluctuations we need to incorporate a customer load model. On
the one hand, the customer load is supposed to be correlated to the grid load.
On the other hand, uncorrelated deviations from customer load forecast and
113
the grid load need to be incorporated. Therefore, we propose to model the
customer load in analogy to the grid load by a load forecast and a SARIMAprocess for short-term deviations (compare Section 3.4.1). In addition to these
components the grid load as a regressor introduces the correlation to the grid
load into the model:
n
o
bt + Xt(C) , a ∈ R.
Ct = exp aLt + C
(7.2)
(C)
In analogy to the model in Equation (3.8) the short term process Xt
is
chosen to be a SARIMA-process with a seasonality of 24 hours and possibly
non-normally distributed innovations. The load forecast consists of a system of
similar-days to capture all seasonalities within the customer load. The model
is chosen to be multiplicative due to the condition of positive load. Especially
when pricing customers with a comparably low load level negative load scenarios are generated by an additive model. This model is a general approach in
order to cover all different types of customers. Therefore, order and parameters
of the SARIMA-process as well as the distribution of the innovations strongly
depend on the customer under consideration. We analyze three customers
representing different classes of customers:
1. A municipal utility delivering households and industrial companies in a
local area.
2. An industrial company ’A’ running its business without any structural
changes.
3. A cyclical industrial company ’B’ being sensitive to the situation of the
economy, i.e. being affected by the crisis in 2009.
Load profiles of these customers for 2008 and 2009 can be found in Figure 7.3.
For comparison these customers are scaled to an average yearly consumption
of 100 GWh. The daily, weekly and yearly patterns of all customers are covbt . Together with the grid
ered by the deterministic component of the model, C
load as a regressor a major part of the customer load can be explained (compare Table 7.2). The intention of including the grid load in the model is the
description of correlations between customer load, grid load and price. As in
Section 7.2.2 stated, all customers are likely to deviate from the load forecast
in the same direction as the price in the same time. Empirical evidence for this
correlation causing a risk for retailers is given in Table 7.3. The importance of
the grid load as a regressor reveals the fact that e.g. the goodness-of-fit of the
model for company B decreases to 0.71 if the grid load is removed from the
model. This is due to B’s strong dependency on the economy. The economic
situation is reflected in the consumption of energy. Therefore the grid load
114
Figure 7.3.: Exemplary load profiles of three customers in the years 2008 till
2009. Top: Municipal utility. Middle: Industrial company A.
Bottom: Industrial company B underlying economical changes.
can explain the decrease in B’s load in 2009 and therefore improve the forecast
quality.
After removing the seasonalities and economical influences by the regression a
stochastic process for the residual time series needs to be fitted. As shown in
Figure 7.4 the residual time series contain autocorrelations with lags of some
hours and some days. These correlations can be covered by a SARIMA-process.
The innovations of these SARIMA-processes are not normally distributed (see
Figure 7.5). In all three cases a Student’s t-distribution with additional scaling
parameter provides a better fit to the empirical data. Empirical work on a wide
range of customers has proven that in case of ’normal’ customers without major
Customer
Municipal utility
Industrial company A
Industrial company B
R2
0.90
0.87
0.86
Table 7.2.: Goodness-of-fit of the customer load model applied to three customers.
115
Customer
Municipal utility
Industrial company A
Industrial company B
Grid load
0.86
0.65
0.72
Spot price
0.61
0.31
0.45
Table 7.3.: Correlations of three customers to grid load and spot price.
changes in demand structure the forecast quality is high. As a consequence, the
stochastic deviations from this forecast are rather small and, more important,
stationary so that the use of a SARIMA-process is justified. Nevertheless, there
are customers increasing the machine park, running bad in their business or
changing structure of their demand for other reasons. These customers will
not satisfy the condition of stationarity as the forecast quality will be low.
Therefore, this model should not be used in these cases. Under the aspect
of risk management, these customers would add a high risk to the retailer’s
portfolio: Structural changes in future times have to be expected but cannot
be forecasted and hedged. Therefore, a higher risk premium is required.
There is no long term process included in the customer load model due to
the lack of customer data on a long term horizon. Long term fluctuations
for economic reasons are indirectly incorporated via the grid load component
which contains a long term component (see Section 3.4.1). In the following we
present an example using the customer load model and the spot price model
introduced in Chapter 3.
7.3.2. Example
Having introduced the model in detail we want to give a practical example
of the model for risk-adequate pricing using the RAROC-approach. Given an
individual customer - see Figure 7.6 for the virtual load and corresponding
scenarios - the model is used for calculation of a risk premium for the hourly
price profile risk, individual volume risk and price-volume correlation.
Following our decomposition of the price of a retail power contract in Section
7.2 we need a basic price as a first step. In this example we derive a basic
price of B = 50.14 EUR/MWh from the HPFC for the year 2013 (compare
Figure 7.1) by means of Equation (7.1). In addition to this price the customer
is charged for the risk premium. Within the calculation of the risk premium
using the RAROC-approach the distribution of costs is needed. We obtain this
(empirical) distribution (see Figure 7.7) as a result of a Monte Carlo simulation
using scenarios from the extended SMaPS model described above.
116
Figure 7.4.: Autocorrelation function for the first 7 days of residual time
series (left) and innovations of the SARIMA-process (right) of
a municipal utility (top), a normal industrial company (middle)
and a cyclical industrial company (bottom).
Apart from the expected costs, M , the RAROC-approach is used for determination of a return on the allocated capital for the risks within the retail
contract. The risk beyond the mean is measured by the 99%-quantile of the
cost distribution and the risk premium consists of a return on the difference
between quantile and mean:
Total price = Expected costs + Hurdle rate · Risk
= M + µ · (99%-quantile − M )
= 50.19 + 0.15 · (51.83 − 50.19) ≈ 50.44
where µ = 0.15 is the assumed hurdle rate. This means the customer within
this example is charged 50.44 EUR/MWh plus an additional sales margin and
risk premiums for further risk factors. Compared to the basic price of 50.14
EUR/MWh a risk premium of 0.30 EUR/MWh is included in the total price.
The mean of the distribution of costs is higher than the basic price due to
expected costs from various risk factors. This result is similar to the one
presented in Burger and Müller (2011) where the SMaPS model is used for the
calculation.
117
Figure 7.5.: Fit of Student’s t-distribution (red) and normal distribution
(green) to kernel density of empirical innovations (blue) for municipal utility (top), industrial company ’A’ (middle) and industrial company ’B’ (bottom) on a logarithmic y-axis.
7.4. Conclusion
From a supplier’s point of view, having the ability to compete on the retail
power market requires the determination of an adequate price for the delivery
of energy within common full-service contracts. Since these contracts include
a high degree of flexibility for customers, the supplier faces two challenges:
1. calculation of a fair price of energy;
2. determination of a premium for the flexibility.
The fair price of energy is deduced from forward contract prices on the wholesale market by breaking down these prices into an hourly granularity. This
step includes estimation of the seasonal structure within the forward contract
intervals. The estimation of price behavior on a holiday day (e.g. Christmas
Eve) is a particularly difficult challenge for the supplier.
118
7.4. Conclusion
Figure 7.6.: A virtual customer load, 1 July 2012 - 31 December 2012, and
two corresponding scenarios, 1 January 2013 - 30 June 2013.
On top of these average costs of delivering energy, the customer is charged for
a risk premium. This premium is the price of the flexibility, which poses a
risk for the supplier. The total risk can be divided into risk factors related to
fluctuations of price and load (such as price validity period, price-volume correlation, individual volume risk, hourly price profile risk and costs of balancing
power) and risks that are not specific for trading on electricity markets (such
as credit risk and operational risk). These risk factors cause expected costs as
well as an increased probability of further costs that threaten the supplier’s
solvency.
Due to the diverse nature of the risk factors, there is no global approach that
covers all risk factors. Therefore, individual approaches have to be applied
to the risk factors in combination with individual risk measures. We have
proposed a correlated price-load model as an extension of the electricity price
model introduced in Chapter 3 to price the hourly price profile risk, pricevolume correlation and individual volume risk simultaneously. These risk factors have the highest potential for the generation of unexpected costs. Using
the load dependent spot price model in combination with the deduction of the
customer load from the grid load by a regression model, a cost distribution
due to these risk factors can be simulated. The application of an adequate
119
Figure 7.7.: The empirical distribution of costs based on 750 correlated price
and load scenarios for year 2013. The basic price, the simulated mean price and the 99%-quantile are the basis for the risk
premium.
risk measure to this distribution leads to the economic capital, which is used
in the RAROC-approach.
The result of calculating an hourly price forward curve and risk premiums for
all mentioned risk factors is a risk-adequate price of a retail power contract.
This price, or this price plus a sales margin, is not necessarily the price on the
retail market. For competitive reasons, deviations from this theoretical price
might be necessary. Nevertheless, risk-adequate pricing is essential for the
supplier in order to gain knowledge about the risks within the retail contracts
and, therefore, the retail portfolio.
120
8. Conclusion
In this thesis we introduce a new model for the spot price of electricity. In this
model, residual load and prices of coal, gas and emission allowances as major
drivers of the electricity price are incorporated. This leads to a model exhibiting the same stylized facts as historical spot prices: seasonalities, spikes, mean
reversion and negative prices. Moreover, the inclusion of load and commodity
prices makes the model applicable to many valuation issues: Electricity derivatives, power plants and retail contracts. To our knowledge it is the first model
including all these factors and, thus, providing a wide range of applications.
Stochastic models for the underlying variables are presented as well. The
residual load model consists of distinct models for grid load and generation
of renewables. As the grid load induces seasonalities to the electricity price
these seasonal variations are covered by the model. Furthermore, economic
load fluctuations, e.g. due to an economic crisis, are considered. Renewables
are modeled by means of a generation forecast resulting from expectations
about future installed capacities and a stochastic process to cover weather
influences.
For the gas price we introduce a spot price model relying on temperature
and oil price as fundamental factors. The temperature is used to derive a
component approximating the influence of the filling level of all gas storages
in the market. In addition to this factor describing the supply side there is the
oil price component reflecting the state of the world economy and influencing
the demand for gas.
Finally, the prices of coal, oil and emission allowances are either modeled by
a three-dimensional random walk or by a correlated two factor model. Both
approaches consider correlations between those commodities and result in correlated non-stationary simulations. As an alternative we introduce a cointegrated model for prices of gas, coal, oil and emission allowances. This model
describes stronger relationships than correlation models for the commodities.
The stationary linear combination of all commodities identified in the cointegrated model suggests the existence of a common trend moving these commodity prices together.
121
8. Conclusion
The modeling framework summarized above is capable of generating realistic
simulation results. The application to pricing of a retail contract gives evidence
for this. Adequate prices for a full-service contract with its various sources of
risk can be derived. Especially, the inclusion of load as an input factor makes
the model applicable to such issues.
A major topic for further research is the inclusion of capacity changes in such
a model. The strong increase of installed capacities of renewables as well as
changes of the conventional power generation mix will lead to further changes
of price behavior. Thus, these effects need to be considered.
The cointegrated model for commodity prices as our preferred model might
be extended as well. So far, only one factor covering short term fluctuations
is considered. By means of a second factor describing cointegrated long term
commodity price behavior the model can be improved. As a consequence a better fit to the expected volatility structure on the market could be achieved.
Altogether, the models introduced in this work provide a good fit to historical
data and, thus, reproduce all relevant features. The resulting simulations are
reliable and can be used for a wide range of applications. With the possible
extensions mentioned above, the model framework is applicable in future as
well.
122
Some of the basic concepts used in the thesis are provided in this chapter. It
is not a comprehensive overview of all methods but gives some information on
the most relevant issues including indications to further literature.
A.1. Stochastic processes
Throughout the thesis we set up time series models in discrete time. All models
contain at least one stochastic process. Definitions and characteristics of these
processes are given in this section. The explanations follow Brockwell and
Davis (1987).
A.1.1. Definitions and properties
Definition A.1 (Stochastic process). A family of random variables {Xt , t ∈
T} on a probability space (Ω, F, P ) is called a stochastic process. The function t 7→ Xt (ω) for ω ∈ Ω is called realization of the stochastic process.
Further notations are path and scenario.
The stochastic processes are observable at any point of time in an index set
T. Processes with a countable index set are called discrete processes. If
the process takes values in R it is denoted as real-valued. As price time series
consist of real-valued prices at certain points in time, we restrict our considerations to discrete, real-valued stochastic processes. The indication of the
index set is omitted as it is usually N or Z. The short notations (Xt ) for the
stochastic process {Xt , t ∈ T} and (xt ) for a realization of the process are used
within this work.
The following definition extends some well-known features of random variables
to stochastic processes.
XV
Definition A.2. Let (Xt ) be a stochastic process.
(i)
µt := E(Xt )
is the mean of (Xt ).
(ii)
σt2 := E (Xt − µt )2 = E(Xt2 ) − µ2t
is the variance of (Xt ).
(iii)
γ(r, s) := E [(Xr − µr ) (Xs − µs )] = E (Xr Xs ) − µr µs
is the autocovariance of (Xt ).
(iv)
ρ(r, s) :=
γ(r, s)
σr σs
is the autocorrelation of (Xt ).
These characteristics are necessary for the definition of stationarity.
Definition A.3 (Stationarity). A stochastic process (Xt ) is called stationary,
if
(i) E|Xt |2 < ∞ for all t,
(ii) µt = m for all t and some m ∈ R and
(iii) γ(r, s) = γ(r + t, s + t) for all t
hold.
Despite different notations in the literature (weak stationarity, covariance stationarity) we refer to this definition as stationarity throughout the thesis.
Definition A.4 (White noise process). A stochastic process (εt ) with
(i) E(εt ) = 0,
(ii) V ar(εt ) = E(ε2t ) = σ 2 and
XVI
(iii) E(εt εs ) = Cov(εt , εs ) = 0,
for all t 6= s,
is called a white noise process, written εt ∼ W N (0, σ 2 ).
Due to this definition, a white noise process is stationary. These definitions are
the basis for the class of autoregressive moving average (ARMA) processes.
Definition A.5 (ARMA process). (Xt ) is called an ARMA process, if (Xt )
is stationary and
p
q
X
X
θj εt−j ,
(A.1)
Xt −
φk Xt−k = εt +
j=1
k=1
holds for all t. εt ∼ W N (0, σ 2 ) and φk , θj ∈ R for all k = 1, . . . , p, j =
1, . . . , q. (p, q) is the order of the process. (εt ) are the innovations of the
process.
The representation in Equation (A.1) is equivalent to
φ(L)Xt = θ(L)εt
(A.2)
where φ(z) = 1 − φ1 z − . . . − φp z p and θ(z) = 1 + θ1 z + . . . + θq z q . The lag
operator L fulfills Lj Xt = Xt−j for j ∈ Z.
By means of the polynomials in Equation (A.2) some important properties for
the parameter estimation can be defined.
Definition A.6. Let (Xt ) with φ(L)Xt = θ(L)εt be an ARMA(p,q) process.
(i) If the polynomials φ(·) and θ(·) have no common zeros and φ(z) 6= 0
holds for all z ∈ C with |z| ≤ 1, then (Xt ) is called causal.
P
P∞
(ii) If πj , j ∈ N, exist, so that ∞
j=0 |πj | < ∞ and εt =
j=0 πj Xt−j holds,
then the ARMA process is called invertible.
Any causal, invertible ARMA process can optionally be formulated as an
AR(∞) or a M A(∞) process. The formulation as a M A(∞) process is based
on Wold’s decomposition.
Theorem A.7 (Wold’s decomposition). A process (Xt ) with variance σ 2 > 0
can be decomposed into
∞
X
Xt =
ψk εt−k + Vt
k=0
where
XVII
(i) ψ0 = 1 and
P∞
k=0
ψj2 < ∞,
(ii) εt ∼ W N (0, σ 2 ),
(iii) εt ∈ Mn = sp {Xt , −∞ < t ≤ n} ∀ t,
(iv) E(Vt εs ) = 0 ∀ t, s,
(v) Vt ∈ M−∞ =
T∞
n=−∞
Mn and
(vi) Vt is a deterministic component.
sp{A} denotes the closed linear span of a set A. Usually the deterministic
component Vt is zero as seasonal components were subtracted from the process
before.
A.1.2. Parameter estimation of ARMA processes
Given a data set and an assumption of the model order the parameters of the
ARMA process need to be estimated. The following method for the estimation
of the ARMA parameters b = (φ1 , . . . , φp , θ1 , . . . , θq )T and the variance of the
white noise process σ 2 is implemented in various software packages.
The parameter estimation for the ARMA process xn = (x1 , . . . , xn )T is based
on the maximization of the likelihood function
1 T −1
2
2 −n/2
−1/2
L(b, σ |xn ) = 2πσ
|Gn (b)|
exp − 2 xn Gn (b)xn .
2σ
This function and the corresponding log-likelihood function
1
n
1
l(b, σ 2 |xn ) = − ln 2πσ 2 − ln (|Gn (b)|) − 2 xTn G−1
n (b)xn
2
2
2σ
(A.3)
rely on the density of the normal distribution. Γn (b) is the covariance matrix
of xn and Gn (b) = σ −2 Γn (b) holds.
From the maximization of l with constant b we obtain
σ
bn2 =
XVIII
1 T −1
x G (b)xn
n n n
as an estimator of the variance of the innovations. As a consequence, we can
rewrite the maximization in Equation (A.3) as the minimization of
1
ln (|Gn (b)|) .
g(b) := ln xTn G−1
n (b)xn /n +
n
Therefore,
b = argmin g(b)
b
b
is an estimator of the parameters of xn . This method is called maximum
likelihood estimation, if the process is assumed to be normally distributed.
The estimator is asymptotically efficient, unbiased and consistent.
These properties of the estimator can be extended to ARMA processes with
non-normally distributed innovations.
Theorem A.8. Let (Xt ) be causal and φ(L)Xt = θ(L)εt an invertible ARMA
process with εt ∼ W N (0, σ 2 ). Then
b → b a.s.
(i) b
(ii) σ
bn2 → σ 2 a.s.
b is asymptotically normally distributed. The parameters depend on b.
(iii) b
This result goes back to Hannan (1973). We use the formulation of Brockwell
and Davis (1987). The proof requires further results including the representation of ARMA processes through spectral densities.
Due to the asymptotical normal distribution the estimator is unbiased and
efficient even in case of non-normally distributed innovations. This method is
denoted as quasi-maximum likelihood estimation.
A.1.3. Test for stationarity
A stationary data set is the basis for the application of an ARMA process. In
order to test for stationarity we introduce the Dickey-Fuller-test (DF-test) and
its augmentation, the Augmented Dickey-Fuller-test (ADF-test) as proposed
by Dickey and Said (1984).
The DF-test is based on the regression approach
Xt = ρXt−1 + εt
XIX
with iid εt ∼ N (0, σ 2 ). A constant term can be included in the approach.
As the data sets within this thesis have zero mean, the constant term is not
considered. The coefficient ρ is estimated by the least squares method
PT
t=1
ρb = P
T
Xt−1 Xt
2
t=1 Xt−1
.
In case ρ = 1 the process is a non-stationary random walk. If |ρ| < 1 holds,
then the time series is stationary. This leads to the hypothesis
H0 : ρ = 1
versus the alternate hypothesis
H1 : |ρ| < 1.
As the test statistics are not valid in the case |ρ| = 1 the differenced situation
∆Xt = (ρ−1)Xt−1 +εt is considered. The test is now stated as the hypothesis
H0 : b = 0
versus the alternate hypothesis
H1 : b < 0
with b = ρ − 1. Dickey and Fuller numerically calculated critical values for the
test statistics of bb.
So far, the method is restricted to AR(1) processes. For many applications
higher orders are needed. A first augmentation is the consideration of AR(p)
processes (p ≥ 2), the ADF-test. Using the lag operator as in Section A.1.1
for the AR(p) process
φ(L)Xt = εt
leads to the hypothesis
H0 : γ := 1 − ρ = 0 versus H1 : −2 < γ < 0
where ρ = φ1 + . . . + φp . The test statistics and critical values are identically
equal to the case p = 1. Dickey and Said (1984) give a further augmentation
to ARMA(p,q) processes for arbitrary (p, q). They show that the same test
statistics are applicable. Tables of critical values are given e.g. by Hamilton
(1994).
XX
A.2. Bessel function
A.2. Bessel function
The differential equation
x2
d2 y
dy
+ (x2 − α2 )y = 0,
+x
2
dx
dx
α ∈ C,
is called Bessel’s differential equation of order α. Usually α ∈ Z holds.
The equation is classified as a second-order linear ordinary differential equation
with two independent solutions. These solutions are called Bessel function
of the first kind, Jα , and Bessel function of the second kind Yα .
If the arguments of the Bessel functions are imaginary, then the Bessel functions are called modified Bessel functions of the first and second kind, Iα
and Kα . These are linear independent solutions of the differential equation
2
2d y
x
dx2
+x
dy
− (x2 − α2 )y = 0,
dx
α ∈ C.
Dependent on the application there are various integral representations for the
modified Bessel function of the second kind, for example
1 1
Kα (x) = e− 2 απi
2
Z
∞
e−ix sinh(t)−αt dt.
−∞
For further details on Bessel functions see Watson (1995) and Abramowitz and
Stegun (1965).
The modified Bessel function of the second kind is also denoted as modified
Bessel function of the third kind, modified Hankel function or MacDonald
function.
A.3. Probability distributions
We give definitions of the most relevant non-standard probability distributions
that are used within this thesis. The definitions can be found in Cont and
Tankov (2004) or Embrechts et al. (2005).
XXI
A.3.1. Student’s t-distribution
Definition A.9 (Student’s t-distribution). A random variable X has a Student’s t-distribution with parameter ν > 0, if its density is
ν+1
2 − 2
Γ ν+1
x
2
1+
f (x) = √
.
ν
νπ Γ ν2
Γ(·) describes the gamma function.
Random numbers can be generated according to the algorithms presented by
Devroye (1986).
A.3.2. Inverse Gaussian distribution
Definition A.10 (Inverse Gaussian distribution). A random variable X has
an inverse Gaussian distribution with parameters λ and µ, if its density
is
!
r
λ (x − µ)2
λ
· exp −
,
x > 0.
f (x) =
2πx3
2µ2 x
An algorithm for generation of random numbers from this distribution is given
by Haas et al. (1976).
A.3.3. Gamma distribution
Definition A.11 (Gamma distribution). A random variable X has a gamma
distribution with parameters a > 0 and b > 0, if its density is
x
1
f (x) = a
xa−1 exp −
,
x > 0.
b Γ(a)
b
We write X ∼ Γ(a, b).
As a second parametrization the use of 1/b instead of b is common.
The gamma distribution has a useful feature for simulation purposes.
b · X ∼ Γ(a, b)
XXII
for X ∼ Γ(a, 1) and b > 0
gives us the possibility to generate Γ(a, 1)-distributed random numbers as a
basis for any other choice of parameters. For this purpose we can use the
rejection method. In case of a ≥ 1 we use a Student’s t-distribution with two
degrees of freedom. In case of 0 < a ≤ 1 the Weibull distribution gives the decision about acceptance or rejection. Detailed descriptions of the corresponding
simulation algorithms are given by Devroye (1986).
A.3.4. Generalized inverse Gaussian distribution
The generalized inverse Gaussian (GIG) distribution is used within the
definition of the generalized hyperbolic distributions in Section A.3.5.
Definition A.12 (GIG distribution). A random variable X with density function
√ λ
χψ
χ−λ
1
λ−1
−1
√ x
exp − χx + ψx ,
x > 0,
f (x) =
2
2Kλ
χψ
follows a GIG distribution with parameters λ, χ, ψ (X ∼ N − (λ, χ, ψ)). Kλ
describes the modified Bessel function of the second kind with index λ (see
Section A.2). The parameters need to satisfy
λ < 0 ⇒ χ > 0, ψ ≥ 0,
λ = 0 ⇒ χ > 0, ψ > 0,
λ > 0 ⇒ χ ≥ 0 ψ > 0.
This definition includes the inverse Gaussian distribution (λ = −0.5) and the
gamma distribution (χ = 0) as special cases. As our applications only rely
on these special cases we do not give a simulation algorithm for the GIG
distribution.
A.3.5. Generalized hyperbolic distributions
The class of generalized hyperbolic (GH) distributions was introduced by
Barndorff-Nielsen (1978). We refer to the parametrization as given by Cont
and Tankov (2004). Prause (1999) gives an overview of further equivalent
parametrizations.
XXIII
Definition A.13 (GH distribution). Let the random variables X ∼ N (0, 1)
and Y ∼ N − (λ, δ 2 , α2 − β 2 ) be independent and
√
d
Z = µ + βY + Y X
the normal mean-variance mixture. Z has a GH distribution with parameters
λ, α, β, δ, µ, written Z ∼ GH(λ, α, β, δ, µ)). Its density function is given by
q
1
λ
2
2 2 − 4
2
Kλ− 1 α δ 2 + (x − µ) eβ(x−µ)
f (x) = C · δ + (x − µ)
2
where
λ/2
C=√
(α2 − β 2 )
p
.
2παλ−1/2 δ λ Kλ δ α2 − β 2
Apart from α > 0 there are further restrictions on the parameters resulting
from the restrictions on the parameters of the GIG distribution (see Section
A.3.4):
λ < 0 ⇒ δ > 0, |β| ≤ α,
λ = 0 ⇒ δ > 0, |β| < α and
λ > 0 ⇒ δ ≥ 0, |β| < α.
Although there are interactions, each parameter has major impact on single
features of the distribution.
λ: shape of the distribution,
α: heaviness of the tails,
β: skewness,
δ: kurtosis,
µ: location.
Amongst others there are some well-known special cases of the GH distribution.
• In case of λ = −0.5 we have the normal inverse Gaussian (NIG) distribution, written N IG(α, β, δ, µ)), with density function
q
2
αδK1 α δ 2 + (x − µ)
p
q
· exp δ α2 − β 2 + β (x − µ) .
f (x) =
π δ 2 + (x − µ)2
XXIV
• If δ = 0, λ > 0, we get
f (x) = √
(α2 − β 2 )
λ−1/2
π (2α)
λ
Γ (λ)
· |x − µ|λ−1/2 · Kλ−1/2 (α|x − µ|) eβ(x−µ) ,
with the gamma function Γ. In this case, f describes the density of the
variance gamma distribution, written V G(λ, α, β, µ)).
• If λ < 0 and α = β = µ = 0, we obtain the well-known Student’s
t-distribution (compare Section A.3.1).
Parameters of any GH distribution can be estimated using the maximum likelihood method.
The simulation algorithm of GH random numbers is the result of the representation as a normal mean-variance mixture. Normal random numbers and GIG
random numbers are required. If NIG or VG random numbers are needed,
the generation of GIG random numbers reduces to the generation of inverse
Gaussian or gamma distributed random numbers. These algorithms are given
in Section A.3.2 and A.3.3.
XXV
B. Programming
The models presented in this work are implemented using the mathematical
software Matlab. As far as available the provided functions were used. We give
a short overview of the most important non-trivial functions used. Detailed
descriptions can be found at MathWorks (2013).
• regress: The function regresses a variable on a set of explaining variables. The coefficients are obtained by a ordinary least squares regression. Confidence intervals as well as some statistical measures are provided. The fundamental components of the spot price models are linear
with respect to the parameters. Therefore, the parameters are estimated
via linear regression.
• mle: This function allows for the maximum likelihood estimation of parameters of a probability distribution. Various standard distributions
including Student’s t-distribution are implemented. The parameters of
user-defined distributions such as truncated distributions can be estimated as well. The truncated Student’s t-distribution including a scale
parameter is fitted via maximum likelihood estimation.
• fminsearch: This function does an unconstrained minimization of a
multivariate function. Various numerical parameters concerning convergence and number of steps can be changed. The algorithm does not
need any numerical or analytical derivatives. Therefore it can be applied
for the optimization problems within this work. This includes the fit
of generalized hyperbolic distributions and the parameter estimation via
Kalman filter.
• adftest: The implementation of the ADF-test for unit roots.
• autocorr: The function provides the autocorrelation function of a time
series. It is used for the determination of model order of the various
SARIMA processes.
• arima: This function of the econometrics toolbox creates a model of
XXVII
B. Programming
the ARIMA class. Using some optional input arguments this function
can define seasonal ARIMA models. The corresponding parameters are
obtained by the function estimate. In addition to the estimation method
the class has a simulation method simulate.
• ksdensity: Given empirical data this function estimates the density
using a smoothing kernel. Various kernels are provided. The kernel
densities plotted within this work are based on the Epanechnikov kernel.
• jcitest: This function tests for cointegration of a set of variables. Test
statistics, critical values, rank of cointegration and parameters are calculated.
Apart from these implemented functions further functions are needed. Especially the load forecasts with similar days require functions dealing with the
calendar. A more complex functionality is required for the estimation of the
multivariate two factor model by Schwartz and Smith (2000). As mentioned
above the Kalman filter is used in this model. As the algorithm was extended
to the multivariate case we implemented the Kalman filter without the corresponding function provided by Matlab.
XXVIII
Bibliography
Abramowitz, M. and Stegun, I. A. (eds.) (1965) Handbook of Mathematical Functions. Dover, New York.
Aïd, R., Campi, L., Huu, A. N., and Touzi, N. (2009) A structural riskneutral model of electricity prices. International Journal of Theoretical and
Applied Finance 12, 925–947.
Alfares, H. K. and Nazeeruddin, M. (2002) Electric load forecasting:
literature survey and classification of methods. International Journal of
Systems Science 33, 23–34.
Anderson, T. W. (1951) Estimating linear restrictions on regression coefficients for multivariate normal distributions. The Annals of Mathematical
Statistics 22, 327–351.
Andersson, G., He, Y., Herzog, F., and Hildmann, M. (2013) Modeling
the merit order curve of the European Energy Exchange power market in
Germany. IEEE Transactions on Power Systems 28, 3155–3164.
Arroyo, J. M., Carrion, M., and Conejo, A. J. (2007) Forward contracting and selling price determination for a retailer. IEEE Transactions
on Power Systems 22, 2105–2114.
Baker, M. P., Mayfield, E. S., and Parsons, J. E. (1998) Alternative
models of uncertain commodity prices for use with modern asset pricing
methods. The Energy Journal 19, 115–148.
Balakrishnan, S., Conejo, A. J., Gabriel, S. A., and Plazas, M. A.
(2006) Optimal price and quantity determination for retail electric power
contracts. IEEE Transactions on Power Systems 21, 180–187.
Balakrishnan, S., Gabriel, S. A., and Sahakij, P. (2004) Optimal retailer load estimates using stochastic dynamic programming. Journal of
Energy Engineering 130, 1–17.
XXIX
Bibliography
Barlow, M. T. (2002) A diffusion model for electricity prices. Mathematical
Finance 12, 287–298.
Barndorff-Nielsen, O. E. (1978) Hyperbolic distributions and distributions on hyperbolae. Scandinavian Journal of Statistics 5, 151–157.
Bartelj, L., Golob, R., Gubina, A. F., and Paravan, D. (2010) Valuating risk from sales contract offer maturity in electricity market. International
Journal of Electrical Power & Energy Systems 32, 147–155.
Benth, F. E. and Benth, J. S. (2007) The volatility of temperature and
pricing of weather derivatives. Quantitative Finance 7, 553–561.
Benth, F. E., Erlwein, C., and Mamon, R. (2010) HMM filtering and
parameter estimation of an electricity spot price model. Energy Economics
32, 1034–1043.
Benth, F. E., Koekkebakker, S., and Ollmar, F. (2007) Extracting and
applying smooth forward curves from average-based commodity contracts
with seasonal variation. The Journal of Derivatives 15, 52–66.
Benz, E. and Trück, S. (2009) Modeling the price dynamics of CO2 emission
allowances. Energy Economics 31, 4–15.
Boogert, A. and de Jong, C. (2008) Gas storage valuation using a Monte
Carlo method. The Journal of Derivatives 15, 81–98.
Boogert, A. and de Jong, C. (2011) Gas storage valuation using a multifactor price process. The Journal of Energy Markets 4, 29–52.
Boroumand, R. and Zachmann, G. (2012) Retailers’ risk management and
vertical arrangements in electricity markets. Energy Policy 40, 465–472.
Brockwell, P. J. and Davis, R. A. (1987) Time series: theory and methods.
Springer-Verlag, New York.
Bunn, D. W., Kettunen, J., and Salo, A. (2010) Optimization of electricity retailer’s contract portfolio subject to risk preferences. IEEE Transactions on Power Systems 25, 117–128.
Bunn, D. W., Kettunen, J., and Salo, A. (2011) Dynamic risk management of electricity contracts with contingent portfolio programming. Management Science submitted manuscript.
XXX
Bibliography
Burger, M., Graeber, B., and Schindlmayr, G. (2007) Managing energy
risk: an integrated view on power and other energy markets. John Wiley &
Sons Ltd, Chichester.
Burger, M., Klar, B., Müller, A., and Schindlmayr, G. (2004) A spot
market model for pricing derivatives in electricity markets. Quantitative
Finance 4, 109–122.
Burger, M. and Müller, J. (2011) Risk-adequate pricing of retail power
contracts. The Journal of Energy Markets 4, 53–75.
Carmona, R. and Coulon, M. (2012) A survey of commodity markets and
structural models for electricity prices. In Benth, F. E. (ed.), Financial Engineering for Energy Asset Management and Hedging in Commodity
Markets. Proceedings from the special thematic year at the Wolfgang Pauli
Institute, Vienna.
Cartea, A. and Figueroa, M. G. (2005) Pricing in electricity markets: a
mean reverting jump diffusion model with seasonality. Applied Mathematical
Finance 12, 313–335.
Cartea, A. and Williams, T. (2008) UK gas markets: the market price
of risk and applications to multiple interruptible supply contracts. Energy
Economics 30, 829–846.
Chen, Z. and Forsyth, P. A. (2010) Implications of a regime-switching
model on natural gas storage valuation and optimal operation. Quantitative
Finance 10, 159–176.
Cont, R. and Tankov, P. (2004) Financial Modelling with Jump Processes.
Chapman & Hall/CRC Press, Boca Raton.
Cortazar, G. and Schwartz, E. S. (2003) Implementing a stochastic model
for oil futures prices. Energy Economics 25, 215–238.
Coulon, M. C. and Howison, S. (2009) Stochastic behaviour of the electricity bid stack: from fundamental drivers to power prices. The Journal of
Energy Markets 2, 29–69.
Daskalakis, G., Markellos, R. N., and Psychoyios, D. (2009) Modeling CO2 emission allowance prices and derivatives: Evidence from the
European trading scheme. Journal of Banking & Finance 33, 1230–1241.
XXXI
Bibliography
Deng, S. (1999) Stochastic models of energy commodity prices and their
applications: mean-reversion with jumps and spikes. Working paper.
Deng, S. and Oren, S. S. (2006) Electricity derivatives and risk management.
Energy 31, 940–953.
Deng, S., Oren, S. S., and Oum, Y. (2005) Volumetric hedging in electricity procurement. In Proceedings of the PowerTech 2005 Conference. St.
Petersburg, Russia.
Devroye, L. (1986) Non-Uniform Random Variate Generation. SpringerVerlag, New York.
Dhaliwal, M., Niimura, T., and Ozawa, K. (2001) Evaluation of retail
electricity supply contracts in deregulated environment. IEEE Power Engineering Society Summer Meeting 2, 1058–1062.
Dickey, D. A. and Said, S. E. (1984) Testing for unit roots in autoregressivemoving average models of unknown order. Biometrika 71, 599–607.
Eakin, K. and Faruqui, A. (2000) Pricing retail electricity: making money
selling a commodity. In Eakin, K. and Faruqui, A. (eds.), Pricing in
Competitive Electricity Markets, chapter 1. Kluwer Academic Publishers.
Elliott, R. J., Sick, G. A., and Stein, M. (2003) Modelling electricity
price risk. Working paper.
Embrechts, P., Frey, R., and McNeil, A. J. (2005) Quantitative risk
management: concepts, techniques, tools. Princeton University Press,
Princeton.
Emery, G. W. and Liu, Q. W. (2002) An analysis of the relationship between
electricity and natural-gas futures prices. The Journal of Futures Markets
22, 95–122.
Engle, R. F. and Granger, C. W. J. (1987) Co-integration and error
correction: representation, estimation, and testing. Econometrica 55, 251–
276.
Escribano, A., Pena, J. I., and Villaplana, P. (2002) Modeling electricity prices: international evidence. Working paper.
XXXII
Bibliography
European
Commission
ets/index_en.htm.
(2013)
www.ec.europa.eu/clima/policies/
European
Energy
www.endextrading.com.
Derivatives
Exchange
(2013)
European Energy Exchange (2013) www.eex.com/en.
European Network of Transmission System Operators for Electricity (2013) www.entsoe.eu.
Eydeland, A. and Wolyniec, K. (2003) Energy and power risk management: new developments in modeling, pricing and hedging. John Wiley &
Sons, Hoboken.
Fanone, E., Gamba, A., and Prokopczuk, M. (2013) The case of negative
day-ahead electricity prices. Energy Economics 35, 22–34.
Fichtner, W., Genoese, M., Keles, D., and Möst, D. (2012) Comparison of extended mean-reversion and time series models for electricity spot
price simulation considering negative prices. Energy Economics 34, 1012–
1032.
Fleten, S.-E. and Lemming, J. (2003) Constructing forward price curves in
electricity markets. Energy Economics 25, 409–424.
Geman, H. (2005) Commodities and commodity derivatives: modeling and
pricing for agriculturals, metals and energy. John Wiley & Sons Ltd, Chichester.
Geman, H. and Roncoroni, A. (2006) Understanding the fine structure of
electricity prices. The Journal of Business 79, 1225–1261.
Goovaerts, M. J., Haezendonck, J., and de Vylder, F. (1984) Insurance premiums: theory and applications. North Holland, Amsterdam.
Granger, C. W. J. (1981) Some properties of time series data and their use
in econometric model specification. Journal of Econometrics 16, 121–130.
Gubina, A., Ilic, M., and Skantze, P. (2000) Bid-based stochastic model
for electricity prices: the impact of fundamental drivers on market dynamics.
Working paper.
XXXIII
Bibliography
Haas, R. W., Michael, J. R., and Schucany, W. R. (1976) Generating
random variates using transformations with multiple roots. The American
Statistician 30, 88–90.
Hallerbach, W. G. (2004) Capital allocation, portfolio enhancement and
performance measurement: a unified approach. In Szegö, G. (ed.), Risk
measures for the 21st century, chapter 20. John Wiley & Sons, Chichester.
Hamilton, J. D. (1994) Time series analysis. Princeton University Press,
Princeton.
Hannan, E. J. (1973) The asymptotic theory of linear time-series models.
Journal of Applied Probability 10, 130–145.
Hatami, A. R., Seifi, H., and Sheikh-El-Eslami, M. K. (2009) Optimal
selling price and energy procurement strategies for a retailer in an electricity
market. Electric Power Systems Research 79, 246–254.
Hirsch, G. (2009) Pricing of hourly exercisable electricity swing options using
different price processes. The Journal of Energy Markets 2, 1–44.
Hirsch, G., Müller, A., and Müller, J. (2013) Modeling the price of
natural gas with temperature and oil price as exogenous factors. Preprint
of University of Siegen.
Horowitz, I., Karimov, R. I., and Woo, C.-K. (2004) Managing electricity
procurement cost and risk by a local distribution company. Energy Policy
32, 635–645.
Huisman, R. and Mahieu, R. (2003) Regime jumps in electricity prices.
Energy Economics 25, 425–434.
IntercontinentalExchange (2013) www.theice.com.
Jaillet, P., Ronn, E. I., and Tompaidis, S. (2004) Valuation of
commodity-based swing options. Management Science 50, 909–921.
Jermakyan, M. and Pirrong, C. (2008) The price of power: the valuation
of power and weather derivatives. Journal of Banking & Finance 32, 2520–
2529.
Johansen, S. (1995) Likelihood-based inference in cointegrated vector autoregressive models. Oxford University Press, Oxford.
XXXIV
Bibliography
de Jong, C. and Schneider, S. (2009) Cointegration between gas and power
spot prices. The Journal of Energy Markets 2, 27–46.
Jorion, P. (2006) Value-at-Risk: the new benchmark for managing financial
risk. McGraw-Hill, New York, 3rd edition.
Kanamura, T. and Ohashi, K. (2007) A structural model for electricity
prices with spikes: measurement of spike risk and optimal policies for hydropower plant operation. Energy Economics 29, 1010–1032.
Karandikar, R. G., Khaparde, S. A., and Kulkarni, S. V. (2007)
Quantifying price risk of electricity retailer based on CAPM and RAROC
methodology. International Journal of Electrical Power & Energy Systems
29, 803–809.
Karandikar, R. G., Khaparde, S. A., and Kulkarni, S. V. (2010)
Strategic evaluation of bilateral contract for electricity retailer in restructured power market. International Journal of Electrical Power & Energy
Systems 32, 457–463.
Keppo, J. and Räsänen, M. (1999) Pricing of electricity tariffs in competitive markets. Energy Economics 21, 213–223.
Koekebakker, S. and Ollmar, F. (2005) Forward curve dynamics in the
nordic electricity market. Managerial Finance 31, 73–94.
Lemming, J. (2003) Risk and investment management in liberalized electricity
markets. Ph.D. thesis, Technical University of Denmark.
Longstaff, F. A. and Schwartz, E. S. (2001) Valuing american options
by simulation: a simple least-squares approach. Review of Financial Studies
14, 113–147.
Lucia, J. J. and Schwartz, E. S. (2002) Electricity prices and power derivatives: evidence from the nordic power exchange. Review of Derivatives Research 5, 5–50.
MathWorks (2013) www.mathworks.com.
Meinshausen, N. and Hambly, B. M. (2004) Monte Carlo methods for the
valuation of multiple-exercise options. Mathematical Finance 14, 557–583.
XXXV
Bibliography
Ojanen, O. J. (2002) Comparative analysis of risk management strategies for
electricity retailers. Master’s thesis, Helsinki University of Technology.
Olsson, M. and Söder, L. (2008) Modeling real-time balancing power market prices using combined SARIMA and Markov processes. IEEE Transactions on Power Systems 23, 443–450.
Paschke, R. and Prokopczuk, M. (2009) Integrating multiple commodities
in a model of stochastic price dynamics. The Journal of Energy Markets 2,
47–82.
Pilipovic, D. (1998) Energy risk: valuing and managing energy derivatives.
McGraw-Hill, New York.
Platts (2013) www.platts.com.
Prause, K. (1999) The generalized hyperbolic model: estimation, financial
derivatives, and risk measures. Ph.D. thesis, University of Freiburg.
Prokopczuk, M., Rachev, S. T., Schindlmayr, G., and Trück, S.
(2007) Quantifying risk in the electricity business: a RAROC-based approach. Energy Economics 29, 1033–1049.
Schneider, S. (2011) Power spot price models with negative prices. The
Journal of Energy Markets 4.
Schwartz, E. S. (1997) The stochastic behaviour of commodity prices: implications for valuation and hedging. The Journal of Finance 52, 923–973.
Schwartz, E. S. and Smith, J. E. (2000) Short-term variations and longterm dynamics in commodity prices. Management Science 46, 893–911.
Seifert, J., Uhrig-Homburg, M., and Wagner, M. (2008) Dynamic behavior of CO2 spot prices. Journal of Environmental Economics and Management 56, 180–194.
Statistisches Bundesamt (2013) www.destatis.de.
Stoll, S.-O. and Wiebauer, K. (2010) A spot price model for natural
gas considering temperature as an exogenous factor and applications. The
Journal of Energy Markets 3, 113–128.
XXXVI
Bibliography
Stoughton, N. M. and Zechner, J. (2007) Optimal capital allocation using
RAROC and EVA. Journal of Financial Intermediation 16, 312–342.
Thoenes, S. (2011) Understanding the determinants of electricity prices and
the impact of the German nuclear moratorium in 2011. Working paper.
Unger, G. (2002) Hedging strategy and electricity contract engineering.
Ph.D. thesis, Swiss Federal Institute of Technology Zurich.
Wagner, A. (2012) Residual demand modeling and application to electricity
pricing. Technical report, Fraunhofer ITWM.
Watson, G. N. (1995) A treatise on the theory of Bessel functions. Cambridge
University Press, 2nd edition.
Weron, R. (2006) Modeling and forecasting electricity loads and prices: a
statistical approach. John Wiley & Sons Ltd, Chichester.
Xu, Z. (2004) Stochastic models for gas prices. Master’s thesis, University of
Calgary.
XXXVII

Stochastic modeling of the spot price of electricity incorporating

Transcrição

Documentos relacionados

LED Collar Light

Economic feasibility for acquisition of efficient refrigerators in

autostop - Altraofficina.it

Christmas Concerts are Coming!

Coal conference in Rio De Janeiro on May 12 to 13 | EnergyAsia +

Schriftliche Prüfungsarbeit zum mittleren

EverPower and Wind Energy in Ohio

Actuarial Mathematics I

ALEXANDRE BRANDÃO Opening: February 19th In his